diff --git a/www/forms-statements.tmpl b/www/forms-statements.tmpl index 7b73a2106cc4cca0a4e7930e55954b83c675d43a..4c26b725b8a333cb7b06d649356c29a01358c4fb 100644 --- a/www/forms-statements.tmpl +++ b/www/forms-statements.tmpl @@ -59,7 +59,7 @@ using the button in the bottom of the page. name="1"> - 1. : The Axiom of Choice: Every set of non-empty sets has a choice function. + 1. : The Axiom of Choice: Every set of non-empty sets has a choice function. @@ -69,7 +69,7 @@ using the button in the bottom of the page. name="2"> - 2. Existence of successor cardinals: For every cardinal there is a cardinal such that and . + 2. Existence of successor cardinals: For every cardinal there is a cardinal such that and . @@ -79,7 +79,7 @@ using the button in the bottom of the page. name="3"> - 3. : For all infinite cardinals , . + 3. : For all infinite cardinals , . @@ -89,7 +89,7 @@ using the button in the bottom of the page. name="4"> - 4. Every infinite set is the union of some disjoint family of denumerable subsets. (Denumerable means .) + 4. Every infinite set is the union of some disjoint family of denumerable subsets. (Denumerable means .) @@ -99,7 +99,7 @@ using the button in the bottom of the page. name="5"> - 5. : Every denumerable set of non-empty denumerable subsets of has a choice function. + 5. : Every denumerable set of non-empty denumerable subsets of has a choice function. @@ -109,7 +109,7 @@ using the button in the bottom of the page. name="6"> - 6. : The union of a denumerable family of denumerable subsets of is denumerable. + 6. : The union of a denumerable family of denumerable subsets of is denumerable. @@ -129,7 +129,7 @@ using the button in the bottom of the page. name="8"> - 8. : Every denumerable family of non-empty sets has a choice function. + 8. : Every denumerable family of non-empty sets has a choice function. @@ -139,7 +139,7 @@ using the button in the bottom of the page. name="9"> - 9. Finite Dedekind finite: (see 1973b): (\ac{Howard/Yorke} \cite{1989}): Every Dedekind finite set is finite. + 9. Finite Dedekind finite: (see 1973b): (\ac{Howard/Yorke} \cite{1989}): Every Dedekind finite set is finite. @@ -149,7 +149,7 @@ using the button in the bottom of the page. name="10"> - 10. : Every denumerable family of non-empty finite sets has a choice function. + 10. : Every denumerable family of non-empty finite sets has a choice function. @@ -159,7 +159,7 @@ using the button in the bottom of the page. name="11"> - 11. A Form of Restricted Choice for Families of Finite Sets: For every infinite set , has an infinite subset such that for every , , the set of all element subsets of has a choice function. + 11. A Form of Restricted Choice for Families of Finite Sets: For every infinite set , has an infinite subset such that for every , , the set of all element subsets of has a choice function. @@ -169,7 +169,7 @@ using the button in the bottom of the page. name="12"> - 12. A Form of Restricted Choice for Families of Finite Sets: For every infinite set and every , there is an infinite subset of such the set of all element subsets of has a choice function. + 12. A Form of Restricted Choice for Families of Finite Sets: For every infinite set and every , there is an infinite subset of such the set of all element subsets of has a choice function. @@ -199,7 +199,7 @@ using the button in the bottom of the page. name="15"> - 15. (KW), The Kinna-Wagner Selection Principle: For every set there is a function such that for all , if then . + 15. (KW), The Kinna-Wagner Selection Principle: For every set there is a function such that for all , if then . @@ -209,7 +209,7 @@ using the button in the bottom of the page. name="16"> - 16. : Every denumerable collection of non-empty sets each with power has a choice function. + 16. : Every denumerable collection of non-empty sets each with power has a choice function. @@ -219,7 +219,7 @@ using the button in the bottom of the page. name="17"> - 17. Ramsey's Theorem I: If is an infinite set and the family of all 2 element subsets of is partitioned into 2 sets and , then there is an infinite subset such that all 2 element subsets of belong to or all 2 element subsets of belong to . (Also, see form 325.) + 17. Ramsey's Theorem I: If is an infinite set and the family of all 2 element subsets of is partitioned into 2 sets and , then there is an infinite subset such that all 2 element subsets of belong to or all 2 element subsets of belong to . (Also, see form 325.) @@ -229,7 +229,7 @@ using the button in the bottom of the page. name="18"> - 18. : The union of a denumerable family of pairwise disjoint pairs has a denumerable subset. + 18. : The union of a denumerable family of pairwise disjoint pairs has a denumerable subset. @@ -249,7 +249,7 @@ using the button in the bottom of the page. name="20"> - 20. If and are families of pairwise disjoint sets and for all , then . + 20. If and are families of pairwise disjoint sets and for all , then . @@ -259,7 +259,7 @@ using the button in the bottom of the page. name="21"> - 21. If is well ordered, and are families of pairwise disjoint sets, and for all , then . + 21. If is well ordered, and are families of pairwise disjoint sets, and for all , then . @@ -269,7 +269,7 @@ using the button in the bottom of the page. name="22"> - 22. : If every member of an infinite set of cardinality has power , then the union has power . + 22. : If every member of an infinite set of cardinality has power , then the union has power . @@ -279,7 +279,7 @@ using the button in the bottom of the page. name="23"> - 23. : For every ordinal , if and every member of has cardinality , then . + 23. : For every ordinal , if and every member of has cardinality , then . @@ -289,7 +289,7 @@ using the button in the bottom of the page. name="24"> - 24. : Every denumerable collection of non-empty sets each with power has a choice function. + 24. : Every denumerable collection of non-empty sets each with power has a choice function. @@ -299,7 +299,7 @@ using the button in the bottom of the page. name="25"> - 25. is regular for all ordinals . + 25. is regular for all ordinals . @@ -309,7 +309,7 @@ using the button in the bottom of the page. name="26"> - 26. : The union of denumerably many sets each of power has power . + 26. : The union of denumerably many sets each of power has power . @@ -319,7 +319,7 @@ using the button in the bottom of the page. name="27"> - 27. : The union of denumerably many sets each of power has power . + 27. : The union of denumerably many sets each of power has power . @@ -329,7 +329,7 @@ using the button in the bottom of the page. name="28"> - 28. (Where is a prime) AL20(): Every vector space over has the property that every linearly independent subset can be extended to a basis. ( is the element field.) + 28. (Where is a prime) AL20(): Every vector space over has the property that every linearly independent subset can be extended to a basis. ( is the element field.) @@ -339,7 +339,7 @@ using the button in the bottom of the page. name="29"> - 29. If and and are families of pairwise disjoint sets and for all , then . + 29. If and and are families of pairwise disjoint sets and for all , then . @@ -359,7 +359,7 @@ using the button in the bottom of the page. name="31"> - 31. : The countable union theorem: The union of a denumerable set of denumerable sets is denumerable. + 31. : The countable union theorem: The union of a denumerable set of denumerable sets is denumerable. @@ -369,7 +369,7 @@ using the button in the bottom of the page. name="32"> - 32. : Every denumerable set of non-empty countable sets has a choice function. + 32. : Every denumerable set of non-empty countable sets has a choice function. @@ -379,7 +379,7 @@ using the button in the bottom of the page. name="33"> - 33. If , : Every linearly ordered set of element sets has a choice function. + 33. If , : Every linearly ordered set of element sets has a choice function. @@ -389,7 +389,7 @@ using the button in the bottom of the page. name="34"> - 34. is regular. + 34. is regular. @@ -409,7 +409,7 @@ using the button in the bottom of the page. name="36"> - 36. Compact T spaces are Loeb. (A space is Loeb if the set of non-empty closed sets has a choice function.) + 36. Compact T spaces are Loeb. (A space is Loeb if the set of non-empty closed sets has a choice function.) @@ -439,7 +439,7 @@ using the button in the bottom of the page. name="39"> - 39. : Every set of non-empty sets such that has a choice function. + 39. : Every set of non-empty sets such that has a choice function. @@ -449,7 +449,7 @@ using the button in the bottom of the page. name="40"> - 40. : Every well orderable set of non-empty sets has a choice function. + 40. : Every well orderable set of non-empty sets has a choice function. @@ -459,7 +459,7 @@ using the button in the bottom of the page. name="41"> - 41. : For every cardinal , or . + 41. : For every cardinal , or . @@ -469,7 +469,7 @@ using the button in the bottom of the page. name="42"> - 42. Löwenheim-Skolem Theorem: If a countable family of first order sentences is satisfiable in a set then it is satisfiable in a countable subset of . + 42. Löwenheim-Skolem Theorem: If a countable family of first order sentences is satisfiable in a set then it is satisfiable in a countable subset of . @@ -479,7 +479,7 @@ using the button in the bottom of the page. name="43"> - 43. (DC), Principle of Dependent Choices: If is a relation on a non-empty set and then there is a sequence of elements of such that . + 43. (DC), Principle of Dependent Choices: If is a relation on a non-empty set and then there is a sequence of elements of such that . @@ -489,7 +489,7 @@ using the button in the bottom of the page. name="44"> - 44. : Given a relation such that for every subset of a set with there is an with , then there is a function such that . + 44. : Given a relation such that for every subset of a set with there is an with , then there is a function such that . @@ -499,7 +499,7 @@ using the button in the bottom of the page. name="45"> - 45. If , : Every set of -element sets has a choice function. + 45. If , : Every set of -element sets has a choice function. @@ -509,7 +509,7 @@ using the button in the bottom of the page. name="46"> - 46. If is a finite subset of , : For every , every set of -element sets has a choice function. + 46. If is a finite subset of , : For every , every set of -element sets has a choice function. @@ -519,7 +519,7 @@ using the button in the bottom of the page. name="47"> - 47. If , : Every well ordered collection of -element sets has a choice function. + 47. If , : Every well ordered collection of -element sets has a choice function. @@ -529,7 +529,7 @@ using the button in the bottom of the page. name="48"> - 48. If is a finite subset of , : For every . + 48. If is a finite subset of , : For every . @@ -549,7 +549,7 @@ using the button in the bottom of the page. name="50"> - 50. Sikorski's Extension Theorem: Every homomorphism of a subalgebra of a Boolean algebra into a complete Boolean algebra can be extended to a homomorphism of into . + 50. Sikorski's Extension Theorem: Every homomorphism of a subalgebra of a Boolean algebra into a complete Boolean algebra can be extended to a homomorphism of into . @@ -569,7 +569,7 @@ using the button in the bottom of the page. name="52"> - 52. Hahn-Banach Theorem: If is a real vector space and satisfies and and is a subspace of and is linear and satisfies then can be extended to such that is linear and . + 52. Hahn-Banach Theorem: If is a real vector space and satisfies and and is a subspace of and is linear and satisfies then can be extended to such that is linear and . @@ -579,7 +579,7 @@ using the button in the bottom of the page. name="53"> - 53. For all infinite cardinals , . + 53. For all infinite cardinals , . @@ -589,7 +589,7 @@ using the button in the bottom of the page. name="54"> - 54. For all infinite cardinals , adj implies is an aleph. ( adj iff and .) + 54. For all infinite cardinals , adj implies is an aleph. ( adj iff and .) @@ -599,7 +599,7 @@ using the button in the bottom of the page. name="55"> - 55. For all infinite cardinals and , if then, or . + 55. For all infinite cardinals and , if then, or . @@ -609,7 +609,7 @@ using the button in the bottom of the page. name="56"> - 56. . ( is Hartogs' aleph, the least not .) + 56. . ( is Hartogs' aleph, the least not .) @@ -619,7 +619,7 @@ using the button in the bottom of the page. name="57"> - 57. If and are Dedekind finite sets then either or . + 57. If and are Dedekind finite sets then either or . @@ -629,7 +629,7 @@ using the button in the bottom of the page. name="58"> - 58. There is an ordinal such that . ( is Hartogs' aleph, the least not .) + 58. There is an ordinal such that . ( is Hartogs' aleph, the least not .) @@ -639,7 +639,7 @@ using the button in the bottom of the page. name="59"> - 59. If is a partial ordering that is not a well ordering, then there is no set such that (the usual injective cardinal ordering on ) is isomorphic to . + 59. If is a partial ordering that is not a well ordering, then there is no set such that (the usual injective cardinal ordering on ) is isomorphic to . @@ -649,7 +649,7 @@ using the button in the bottom of the page. name="60"> - 60. : Every set of non-empty, well orderable sets has a choice function. + 60. : Every set of non-empty, well orderable sets has a choice function. @@ -659,7 +659,7 @@ using the button in the bottom of the page. name="61"> - 61. ): For each , , every set of element sets has a choice function. + 61. ): For each , , every set of element sets has a choice function. @@ -669,7 +669,7 @@ using the button in the bottom of the page. name="62"> - 62. : Every set of non-empty finite sets has a choice function. + 62. : Every set of non-empty finite sets has a choice function. @@ -679,7 +679,7 @@ using the button in the bottom of the page. name="63"> - 63. : Weak ultrafilter principle: Every infinite set has a non-trivial ultrafilter. + 63. : Weak ultrafilter principle: Every infinite set has a non-trivial ultrafilter. @@ -689,7 +689,7 @@ using the button in the bottom of the page. name="64"> - 64. (see 1989): There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.) + 64. (see 1989): There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.) @@ -699,7 +699,7 @@ using the button in the bottom of the page. name="65"> - 65. The Krein-Milman Theorem: Let be a compact convex set in a locally convex topological vector space . Then has an extreme point. (An extreme point is a point which is not an interior point of any line segment which lies in .) + 65. The Krein-Milman Theorem: Let be a compact convex set in a locally convex topological vector space . Then has an extreme point. (An extreme point is a point which is not an interior point of any line segment which lies in .) @@ -719,7 +719,7 @@ using the button in the bottom of the page. name="67"> - 67. (MC), The Axiom of Multiple Choice: For every set of non-empty sets there is a function such that and is finite). + 67. (MC), The Axiom of Multiple Choice: For every set of non-empty sets there is a function such that and is finite). @@ -749,7 +749,7 @@ using the button in the bottom of the page. name="70"> - 70. There is a non-trivial ultrafilter on . + 70. There is a non-trivial ultrafilter on . @@ -759,7 +759,7 @@ using the button in the bottom of the page. name="71"> - 71. : or . + 71. : or . @@ -769,7 +769,7 @@ using the button in the bottom of the page. name="72"> - 72. Artin-Schreier Theorem: Every field in which is not the sum of squares can be ordered. (The ordering, , must satisfy (a) for all and (b) and c.) + 72. Artin-Schreier Theorem: Every field in which is not the sum of squares can be ordered. (The ordering, , must satisfy (a) for all and (b) and c.) @@ -779,7 +779,7 @@ using the button in the bottom of the page. name="73"> - 73. , : For every , if is an infinite family of element sets, then has an infinite subfamily with a choice function. + 73. , : For every , if is an infinite family of element sets, then has an infinite subfamily with a choice function. @@ -789,7 +789,7 @@ using the button in the bottom of the page. name="74"> - 74. For every the following are equivalent: (1) is closed and bounded. \itemitem{(2)} Every sequence has a convergent subsequence with limit in A. + 74. For every the following are equivalent: (1) is closed and bounded. \itemitem{(2)} Every sequence has a convergent subsequence with limit in A. @@ -809,7 +809,7 @@ using the button in the bottom of the page. name="76"> - 76. (-MC): For every family of pairwise disjoint non-empty sets, there is a function such that for each , f(x) is a non-empty countable subset of . + 76. (-MC): For every family of pairwise disjoint non-empty sets, there is a function such that for each , f(x) is a non-empty countable subset of . @@ -819,7 +819,7 @@ using the button in the bottom of the page. name="77"> - 77. A linear ordering of a set is a well ordering if and only if has no infinite descending sequences. + 77. A linear ordering of a set is a well ordering if and only if has no infinite descending sequences. @@ -829,7 +829,7 @@ using the button in the bottom of the page. name="78"> - 78. Urysohn's Lemma: If and are disjoint closed sets in a normal space , then there is a continuous which is 1 everywhere in and 0 everywhere in . + 78. Urysohn's Lemma: If and are disjoint closed sets in a normal space , then there is a continuous which is 1 everywhere in and 0 everywhere in . @@ -849,7 +849,7 @@ using the button in the bottom of the page. name="80"> - 80. : Every denumerable set of pairs has a choice function. + 80. : Every denumerable set of pairs has a choice function. @@ -859,7 +859,7 @@ using the button in the bottom of the page. name="81"> - 81. (For ) : For every set there is an ordinal and a one to one function . ( and . ( is equivalent to form 1 (AC) and is equivalent to the selection principle (form 15)). + 81. (For ) : For every set there is an ordinal and a one to one function . ( and . ( is equivalent to form 1 (AC) and is equivalent to the selection principle (form 15)). @@ -869,7 +869,7 @@ using the button in the bottom of the page. name="82"> - 82. (see 1989): If is infinite then is Dedekind infinite. ( is finite is Dedekind finite.) + 82. (see 1989): If is infinite then is Dedekind infinite. ( is finite is Dedekind finite.) @@ -879,7 +879,7 @@ using the button in the bottom of the page. name="83"> - 83. (see 1989): -finite is equivalent to finite. + 83. (see 1989): -finite is equivalent to finite. @@ -889,7 +889,7 @@ using the button in the bottom of the page. name="84"> - 84. (see 1989): is -finite if and only if is Dedekind finite). + 84. (see 1989): is -finite if and only if is Dedekind finite). @@ -899,7 +899,7 @@ using the button in the bottom of the page. name="85"> - 85. : Every family of denumerable sets has a choice function. + 85. : Every family of denumerable sets has a choice function. @@ -909,7 +909,7 @@ using the button in the bottom of the page. name="86"> - 86. : If is a set of non-empty sets such that , then has a choice function. + 86. : If is a set of non-empty sets such that , then has a choice function. @@ -919,7 +919,7 @@ using the button in the bottom of the page. name="87"> - 87. : Given a relation such that for every subset of a set with , there is an with then there is a function such that () . + 87. : Given a relation such that for every subset of a set with , there is an with then there is a function such that () . @@ -929,7 +929,7 @@ using the button in the bottom of the page. name="88"> - 88. : Every family of pairs has a choice function. + 88. : Every family of pairs has a choice function. @@ -949,7 +949,7 @@ using the button in the bottom of the page. name="90"> - 90. : Every linearly ordered set can be well ordered. + 90. : Every linearly ordered set can be well ordered. @@ -959,7 +959,7 @@ using the button in the bottom of the page. name="91"> - 91. : The power set of a well ordered set can be well ordered. + 91. : The power set of a well ordered set can be well ordered. @@ -969,7 +969,7 @@ using the button in the bottom of the page. name="92"> - 92. : Every well ordered family of non-empty subsets of has a choice function. + 92. : Every well ordered family of non-empty subsets of has a choice function. @@ -989,7 +989,7 @@ using the button in the bottom of the page. name="94"> - 94. : Every denumerable family of non-empty sets of reals has a choice function. + 94. : Every denumerable family of non-empty sets of reals has a choice function. @@ -999,7 +999,7 @@ using the button in the bottom of the page. name="95"> - 95. Existence of Complementary Subspaces over a Field : If is a field, then every vector space over has the property that if is a subspace of , then there is a subspace such that and generates . + 95. Existence of Complementary Subspaces over a Field : If is a field, then every vector space over has the property that if is a subspace of , then there is a subspace such that and generates . @@ -1009,7 +1009,7 @@ using the button in the bottom of the page. name="96"> - 96. Löwig's Theorem. If and are both bases for the vector space then . + 96. Löwig's Theorem. If and are both bases for the vector space then . @@ -1019,7 +1019,7 @@ using the button in the bottom of the page. name="97"> - 97. Cardinal Representatives: For every set there is a function with domain such that for all , (i) and (ii) . + 97. Cardinal Representatives: For every set there is a function with domain such that for all , (i) and (ii) . @@ -1039,7 +1039,7 @@ using the button in the bottom of the page. name="99"> - 99. Rado's Selection Lemma: Let be a family of finite subsets (of ) and suppose for each finite there is a function such that . Then there is an such that for every finite there is a finite such that and such that and agree on S. + 99. Rado's Selection Lemma: Let be a family of finite subsets (of ) and suppose for each finite there is a function such that . Then there is an such that for every finite there is a finite such that and such that and agree on S. @@ -1049,7 +1049,7 @@ using the button in the bottom of the page. name="100"> - 100. Weak Partition Principle: For all sets and , if , then it is not the case that . + 100. Weak Partition Principle: For all sets and , if , then it is not the case that . @@ -1059,7 +1059,7 @@ using the button in the bottom of the page. name="101"> - 101. Partition Principle: If is a partition of , then . + 101. Partition Principle: If is a partition of , then . @@ -1069,7 +1069,7 @@ using the button in the bottom of the page. name="102"> - 102. For all Dedekind finite cardinals and , if then . + 102. For all Dedekind finite cardinals and , if then . @@ -1079,7 +1079,7 @@ using the button in the bottom of the page. name="103"> - 103. If is a linear ordering and then some initial segment of is uncountable. + 103. If is a linear ordering and then some initial segment of is uncountable. @@ -1099,7 +1099,7 @@ using the button in the bottom of the page. name="105"> - 105. There is a partially ordered set such that for no set is (the ordering on is the usual injective cardinal ordering) isomorphic to . + 105. There is a partially ordered set such that for no set is (the ordering on is the usual injective cardinal ordering) isomorphic to . @@ -1119,7 +1119,7 @@ using the button in the bottom of the page. name="107"> - 107. M.~Hall's Theorem: Let be a collection of finite subsets (of a set ) then if for each finite there is an injective choice function on () then there is an injective choice function on . (That is, a 1-1 function such that .) (According to a theorem of P.~Hall () is equivalent to . P.~Hall's theorem does not require the axiom of choice.) + 107. M.~Hall's Theorem: Let be a collection of finite subsets (of a set ) then if for each finite there is an injective choice function on () then there is an injective choice function on . (That is, a 1-1 function such that .) (According to a theorem of P.~Hall () is equivalent to . P.~Hall's theorem does not require the axiom of choice.) @@ -1129,7 +1129,7 @@ using the button in the bottom of the page. name="108"> - 108. There is an ordinal such that is not the union of a denumerable set of denumerable sets. + 108. There is an ordinal such that is not the union of a denumerable set of denumerable sets. @@ -1139,7 +1139,7 @@ using the button in the bottom of the page. name="109"> - 109. Every field and every vector space over has the property that each linearly independent set can be extended to a basis. + 109. Every field and every vector space over has the property that each linearly independent set can be extended to a basis. @@ -1149,7 +1149,7 @@ using the button in the bottom of the page. name="110"> - 110. Every vector space over has a basis. + 110. Every vector space over has a basis. @@ -1159,7 +1159,7 @@ using the button in the bottom of the page. name="111"> - 111. : The union of an infinite well ordered set of 2-element sets is an infinite well ordered set. + 111. : The union of an infinite well ordered set of 2-element sets is an infinite well ordered set. @@ -1169,7 +1169,7 @@ using the button in the bottom of the page. name="112"> - 112. : For every family of non-empty sets each of which can be linearly ordered there is a function such that for all , is a non-empty finite subset of . + 112. : For every family of non-empty sets each of which can be linearly ordered there is a function such that for all , is a non-empty finite subset of . @@ -1189,7 +1189,7 @@ using the button in the bottom of the page. name="114"> - 114. Every A-bounded topological space is weakly Loeb. ( A-bounded means amorphous subsets are relatively compact. {\it Weakly Loeb} means the set of non-empty closed subsets has a multiple choice function.) + 114. Every A-bounded topological space is weakly Loeb. ( A-bounded means amorphous subsets are relatively compact. {\it Weakly Loeb} means the set of non-empty closed subsets has a multiple choice function.) @@ -1199,7 +1199,7 @@ using the button in the bottom of the page. name="115"> - 115. The product of weakly Loeb spaces is weakly Loeb. + 115. The product of weakly Loeb spaces is weakly Loeb. @@ -1209,7 +1209,7 @@ using the button in the bottom of the page. name="116"> - 116. Every compact space is weakly Loeb. + 116. Every compact space is weakly Loeb. @@ -1219,7 +1219,7 @@ using the button in the bottom of the page. name="117"> - 117. If is a measurable cardinal, then is the th inaccessible cardinal. + 117. If is a measurable cardinal, then is the th inaccessible cardinal. @@ -1239,7 +1239,7 @@ using the button in the bottom of the page. name="119"> - 119. ,uniformly orderable with order type of the integers): Suppose is a set and there is a function such that for each is an ordering of of type (the usual ordering of the integers), then has a choice function. + 119. ,uniformly orderable with order type of the integers): Suppose is a set and there is a function such that for each is an ordering of of type (the usual ordering of the integers), then has a choice function. @@ -1249,7 +1249,7 @@ using the button in the bottom of the page. name="120"> - 120. If , : Every linearly ordered set of non-empty sets each of whose cardinality is in has a choice function. + 120. If , : Every linearly ordered set of non-empty sets each of whose cardinality is in has a choice function. @@ -1259,7 +1259,7 @@ using the button in the bottom of the page. name="121"> - 121. : Every linearly ordered set of non-empty finite sets has a choice function. + 121. : Every linearly ordered set of non-empty finite sets has a choice function. @@ -1269,7 +1269,7 @@ using the button in the bottom of the page. name="122"> - 122. : Every well ordered set of non-empty finite sets has a choice function. + 122. : Every well ordered set of non-empty finite sets has a choice function. @@ -1279,7 +1279,7 @@ using the button in the bottom of the page. name="123"> - 123. : Uniform weak ultrafilter principle: For each family of infinite sets such that , is a non-principal ultrafilter on . + 123. : Uniform weak ultrafilter principle: For each family of infinite sets such that , is a non-principal ultrafilter on . @@ -1299,7 +1299,7 @@ using the button in the bottom of the page. name="125"> - 125. There does not exist an infinite, compact connected space. (A space is a space in which the intersection of any well orderable family of open sets is open.) + 125. There does not exist an infinite, compact connected space. (A space is a space in which the intersection of any well orderable family of open sets is open.) @@ -1309,7 +1309,7 @@ using the button in the bottom of the page. name="126"> - 126. , Countable axiom of multiple choice: For every denumerable set of non-empty sets there is a function such that for all , is a non-empty finite subset of . + 126. , Countable axiom of multiple choice: For every denumerable set of non-empty sets there is a function such that for all , is a non-empty finite subset of . @@ -1319,7 +1319,7 @@ using the button in the bottom of the page. name="127"> - 127. An amorphous power of a compact space, which as a set is well orderable, is well orderable. + 127. An amorphous power of a compact space, which as a set is well orderable, is well orderable. @@ -1339,7 +1339,7 @@ using the button in the bottom of the page. name="129"> - 129. For every infinite set , admits a partition into sets of order type . (For every infinite , there is a set such that is a partition of and for each , is an ordering of of type .) + 129. For every infinite set , admits a partition into sets of order type . (For every infinite , there is a set such that is a partition of and for each , is an ordering of of type .) @@ -1349,7 +1349,7 @@ using the button in the bottom of the page. name="130"> - 130. is well orderable. + 130. is well orderable. @@ -1359,7 +1359,7 @@ using the button in the bottom of the page. name="131"> - 131. : For every denumerable family of pairwise disjoint non-empty sets, there is a function such that for each , f(x) is a non-empty countable subset of . + 131. : For every denumerable family of pairwise disjoint non-empty sets, there is a function such that for each , f(x) is a non-empty countable subset of . @@ -1369,7 +1369,7 @@ using the button in the bottom of the page. name="132"> - 132. : Every infinite family of finite sets has an infinite subfamily with a choice function. + 132. : Every infinite family of finite sets has an infinite subfamily with a choice function. @@ -1389,7 +1389,7 @@ using the button in the bottom of the page. name="134"> - 134. If is an infinite space and is , then is countable. ( is hereditarily ''.) + 134. If is an infinite space and is , then is countable. ( is hereditarily ''.) @@ -1399,7 +1399,7 @@ using the button in the bottom of the page. name="135"> - 135. If is a space with at least two points and is hereditarily metacompact then is countable. (A space is metacompact if every open cover has an open point finite refinement. If and are covers of a space , then is a {\it refinement} of if . is {\it point finite} if there are only finitely many such that .) + 135. If is a space with at least two points and is hereditarily metacompact then is countable. (A space is metacompact if every open cover has an open point finite refinement. If and are covers of a space , then is a {\it refinement} of if . is {\it point finite} if there are only finitely many such that .) @@ -1409,7 +1409,7 @@ using the button in the bottom of the page. name="136"> - 136. Surjective Cardinal Cancellation (depends on ): For all cardinals and , implies . + 136. Surjective Cardinal Cancellation (depends on ): For all cardinals and , implies . @@ -1419,7 +1419,7 @@ using the button in the bottom of the page. name="137"> - 137. Suppose . If is a 1-1 map from into then there are partitions and of and such that maps onto . + 137. Suppose . If is a 1-1 map from into then there are partitions and of and such that maps onto . @@ -1429,7 +1429,7 @@ using the button in the bottom of the page. name="138"> - 138. Suppose . If is a partial map from onto (that is, the domain is a subset of ), then there are partitions and of and such that maps onto . + 138. Suppose . If is a partial map from onto (that is, the domain is a subset of ), then there are partitions and of and such that maps onto . @@ -1439,7 +1439,7 @@ using the button in the bottom of the page. name="139"> - 139. Using the discrete topology on 2, is compact. + 139. Using the discrete topology on 2, is compact. @@ -1449,7 +1449,7 @@ using the button in the bottom of the page. name="140"> - 140. Let be the set of all (undirected) infinite cycles of reals (Graphs whose vertices are real numbers, connected, no loops and each vertex adjacent to exactly two others). Then there is a function on such that for all , is a direction along . + 140. Let be the set of all (undirected) infinite cycles of reals (Graphs whose vertices are real numbers, connected, no loops and each vertex adjacent to exactly two others). Then there is a function on such that for all , is a direction along . @@ -1459,7 +1459,7 @@ using the button in the bottom of the page. name="141"> - 141. [14 P()] with : Let be a collection of sets such that and suppose is a symmetric binary relation on such that for all finite there is an consistent choice function for . Then there is an consistent choice function for . + 141. [14 P()] with : Let be a collection of sets such that and suppose is a symmetric binary relation on such that for all finite there is an consistent choice function for . Then there is an consistent choice function for . @@ -1469,7 +1469,7 @@ using the button in the bottom of the page. name="142"> - 142. : There is a set of reals without the property of Baire. + 142. : There is a set of reals without the property of Baire. @@ -1479,7 +1479,7 @@ using the button in the bottom of the page. name="143"> - 143. : If is a connected relation ( or ) then contains a -maximal transitive subset. + 143. : If is a connected relation ( or ) then contains a -maximal transitive subset. @@ -1499,7 +1499,7 @@ using the button in the bottom of the page. name="145"> - 145. Compact -spaces are Dedekind finite. (A -space is a topological space in which the intersection of a countable collection of open sets is open.) + 145. Compact -spaces are Dedekind finite. (A -space is a topological space in which the intersection of a countable collection of open sets is open.) @@ -1509,7 +1509,7 @@ using the button in the bottom of the page. name="146"> - 146. : For every topological space , if is a continuous finite to one image of an A1 space then is an A1 space. ( is A1 means if covers then such that + 146. : For every topological space , if is a continuous finite to one image of an A1 space then is an A1 space. ( is A1 means if covers then such that @@ -1519,7 +1519,7 @@ using the button in the bottom of the page. name="147"> - 147. : Every topological space can be covered by a well ordered family of discrete sets. + 147. : Every topological space can be covered by a well ordered family of discrete sets. @@ -1529,7 +1529,7 @@ using the button in the bottom of the page. name="148"> - 148. : For every topological space , if is well ordered, then has a well ordered base. + 148. : For every topological space , if is well ordered, then has a well ordered base. @@ -1539,7 +1539,7 @@ using the button in the bottom of the page. name="149"> - 149. : Every topological space is a continuous, finite to one image of an A1 space. + 149. : Every topological space is a continuous, finite to one image of an A1 space. @@ -1549,7 +1549,7 @@ using the button in the bottom of the page. name="150"> - 150. : Every infinite set of denumerable sets has an infinite subset with a choice function. + 150. : Every infinite set of denumerable sets has an infinite subset with a choice function. @@ -1559,7 +1559,7 @@ using the button in the bottom of the page. name="151"> - 151. (): The union of a well ordered set of denumerable sets is well orderable. + 151. (): The union of a well ordered set of denumerable sets is well orderable. @@ -1569,7 +1569,7 @@ using the button in the bottom of the page. name="152"> - 152. : Every non-well-orderable set is the union of a pairwise disjoint, well orderable family of denumerable sets. + 152. : Every non-well-orderable set is the union of a pairwise disjoint, well orderable family of denumerable sets. @@ -1589,7 +1589,7 @@ using the button in the bottom of the page. name="154"> - 154. Tychonoff's Compactness Theorem for Countably Many Spaces: The product of countably many compact spaces is compact. + 154. Tychonoff's Compactness Theorem for Countably Many Spaces: The product of countably many compact spaces is compact. @@ -1599,7 +1599,7 @@ using the button in the bottom of the page. name="155"> - 155. : There are no non-trivial Läuchli continua. (A Läuchli continuum is a strongly connected continuum. {\it Continuum} compact, connected, Hausdorff space; and {\it strongly connected} every continuous real valued function is constant.) + 155. : There are no non-trivial Läuchli continua. (A Läuchli continuum is a strongly connected continuum. {\it Continuum} compact, connected, Hausdorff space; and {\it strongly connected} every continuous real valued function is constant.) @@ -1609,7 +1609,7 @@ using the button in the bottom of the page. name="156"> - 156. Theorem of Gelfand and Kolmogoroff: Two compact spaces are homeomorphic if their rings of real valued continuous functions are isomorphic. + 156. Theorem of Gelfand and Kolmogoroff: Two compact spaces are homeomorphic if their rings of real valued continuous functions are isomorphic. @@ -1619,7 +1619,7 @@ using the button in the bottom of the page. name="157"> - 157. Theorem of Goodner: A compact space is extremally disconnected (the closure of every open set is open) if and only if each non-empty subset of (set of continuous real valued functions on ) which is pointwise bounded has a supremum. + 157. Theorem of Goodner: A compact space is extremally disconnected (the closure of every open set is open) if and only if each non-empty subset of (set of continuous real valued functions on ) which is pointwise bounded has a supremum. @@ -1629,7 +1629,7 @@ using the button in the bottom of the page. name="158"> - 158. In every Hilbert space , if the closed unit ball is sequentially compact, then has an orthonormal basis. + 158. In every Hilbert space , if the closed unit ball is sequentially compact, then has an orthonormal basis. @@ -1659,7 +1659,7 @@ using the button in the bottom of the page. name="161"> - 161. Definability of cardinal addition in terms of : There is a first order formula whose only non-logical symbol is (for cardinals) that defines cardinal addition. + 161. Definability of cardinal addition in terms of : There is a first order formula whose only non-logical symbol is (for cardinals) that defines cardinal addition. @@ -1669,7 +1669,7 @@ using the button in the bottom of the page. name="162"> - 162. Non-existence of infinite units: There is no infinite cardinal number such that and for all cardinals and , or . + 162. Non-existence of infinite units: There is no infinite cardinal number such that and for all cardinals and , or . @@ -1699,7 +1699,7 @@ using the button in the bottom of the page. name="165"> - 165. : Every well ordered family of non-empty, well orderable sets has a choice function. + 165. : Every well ordered family of non-empty, well orderable sets has a choice function. @@ -1709,7 +1709,7 @@ using the button in the bottom of the page. name="166"> - 166. : Every infinite family of pairs has an infinite subfamily with a choice function. + 166. : Every infinite family of pairs has an infinite subfamily with a choice function. @@ -1719,7 +1719,7 @@ using the button in the bottom of the page. name="167"> - 167. , Partial Kinna-Wagner Principle: For every denumerable family such that for all , , there is an infinite subset and a function such that for all , . + 167. , Partial Kinna-Wagner Principle: For every denumerable family such that for all , , there is an infinite subset and a function such that for all , . @@ -1729,7 +1729,7 @@ using the button in the bottom of the page. name="168"> - 168. Dual Cantor-Bernstein Theorem: and implies . + 168. Dual Cantor-Bernstein Theorem: and implies . @@ -1749,7 +1749,7 @@ using the button in the bottom of the page. name="170"> - 170. . + 170. . @@ -1759,7 +1759,7 @@ using the button in the bottom of the page. name="171"> - 171. If is a partial order such that is the denumerable union of finite sets and all antichains in are finite then for each denumerable family of dense sets there is a generic filter. + 171. If is a partial order such that is the denumerable union of finite sets and all antichains in are finite then for each denumerable family of dense sets there is a generic filter. @@ -1769,7 +1769,7 @@ using the button in the bottom of the page. name="172"> - 172. For every infinite set , if is hereditarily countable (that is, every is countable) then . + 172. For every infinite set , if is hereditarily countable (that is, every is countable) then . @@ -1789,7 +1789,7 @@ using the button in the bottom of the page. name="174"> - 174. : The representation theorem for multi-algebras with unary operations: Assume is a multi-algebra with unary operations (and no other operations). Then there is an algebra with unary operations and an equivalence relation on such that and are isomorphic multi-algebras. + 174. : The representation theorem for multi-algebras with unary operations: Assume is a multi-algebra with unary operations (and no other operations). Then there is an algebra with unary operations and an equivalence relation on such that and are isomorphic multi-algebras. @@ -1799,7 +1799,7 @@ using the button in the bottom of the page. name="175"> - 175. Transitivity Condition: For all sets , there is a set abd a function such that is transitive and is a one to one function from onto . von + 175. Transitivity Condition: For all sets , there is a set abd a function such that is transitive and is a one to one function from onto . von @@ -1819,7 +1819,7 @@ using the button in the bottom of the page. name="177"> - 177. An infinite box product of regular spaces, each of cardinality greater than 1, is neither first countable nor connected. + 177. An infinite box product of regular spaces, each of cardinality greater than 1, is neither first countable nor connected. @@ -1829,7 +1829,7 @@ using the button in the bottom of the page. name="178"> - 178. If , and , , : If is any set of -element sets then there is a function with domain such that for all , and . + 178. If , and , , : If is any set of -element sets then there is a function with domain such that for all , and . @@ -1839,7 +1839,7 @@ using the button in the bottom of the page. name="179"> - 179. Suppose is an ordinal. , ). + 179. Suppose is an ordinal. , ). @@ -1849,7 +1849,7 @@ using the button in the bottom of the page. name="180"> - 180. Every Abelian group has a divisible hull. (If and are groups, is a divisible hull of means is a divisible group, is a subgroup of and for every non-zero , such that .) + 180. Every Abelian group has a divisible hull. (If and are groups, is a divisible hull of means is a divisible group, is a subgroup of and for every non-zero , such that .) @@ -1859,7 +1859,7 @@ using the button in the bottom of the page. name="181"> - 181. : Every set of non-empty sets such that has a choice function. + 181. : Every set of non-empty sets such that has a choice function. @@ -1869,7 +1869,7 @@ using the button in the bottom of the page. name="182"> - 182. There is an aleph whose cofinality is greater than . + 182. There is an aleph whose cofinality is greater than . @@ -1879,7 +1879,7 @@ using the button in the bottom of the page. name="183"> - 183. There are no minimal sets. That is, there are no sets such that (1) is incomparable with \itemitem{(2)} for every and \itemitem{(3)} or . + 183. There are no minimal sets. That is, there are no sets such that (1) is incomparable with \itemitem{(2)} for every and \itemitem{(3)} or . @@ -1889,7 +1889,7 @@ using the button in the bottom of the page. name="184"> - 184. Existence of a double uniformization: For all and , for all , if there is an infinite cardinal satisfying: (1) , and \itemitem{(2)} , , then such that for all such that and such that . ( is called a double uniformization of .) + 184. Existence of a double uniformization: For all and , for all , if there is an infinite cardinal satisfying: (1) , and \itemitem{(2)} , , then such that for all such that and such that . ( is called a double uniformization of .) @@ -1929,7 +1929,7 @@ using the button in the bottom of the page. name="188"> - 188. : For every Abelian group there is a projective Abelian group and a homomorphism from onto . + 188. : For every Abelian group there is a projective Abelian group and a homomorphism from onto . @@ -1939,7 +1939,7 @@ using the button in the bottom of the page. name="189"> - 189. : For every Abelian group there is an injective Abelian group and a one to one homomorphism from into . + 189. : For every Abelian group there is an injective Abelian group and a one to one homomorphism from into . @@ -1959,7 +1959,7 @@ using the button in the bottom of the page. name="191"> - 191. : There is a set such that for every set , there is an ordinal and a function from onto . + 191. : There is a set such that for every set , there is an ordinal and a function from onto . @@ -1969,7 +1969,7 @@ using the button in the bottom of the page. name="192"> - 192. sets: For every set there is a projective set and a function from onto . + 192. sets: For every set there is a projective set and a function from onto . @@ -1979,7 +1979,7 @@ using the button in the bottom of the page. name="193"> - 193. : Every Abelian group is a homomorphic image of a free projective Abelian group. + 193. : Every Abelian group is a homomorphic image of a free projective Abelian group. @@ -1989,7 +1989,7 @@ using the button in the bottom of the page. name="194"> - 194. or : If , has domain , and is in , then there is a sequence of elements of with for all . + 194. or : If , has domain , and is in , then there is a sequence of elements of with for all . @@ -2009,7 +2009,7 @@ using the button in the bottom of the page. name="196"> - 196. and are not both measurable. + 196. and are not both measurable. @@ -2019,7 +2019,7 @@ using the button in the bottom of the page. name="197"> - 197. is the union of three sets with the property that for all there is a straight line such that . + 197. is the union of three sets with the property that for all there is a straight line such that . @@ -2029,7 +2029,7 @@ using the button in the bottom of the page. name="198"> - 198. For every set , if the only linearly orderable subsets of are the finite subsets of , then either is finite or has an amorphous subset. + 198. For every set , if the only linearly orderable subsets of are the finite subsets of , then either is finite or has an amorphous subset. @@ -2039,7 +2039,7 @@ using the button in the bottom of the page. name="199"> - 199. (For ) If all , Dedekind finite subsets of are finite, then all Dedekind finite subsets of are finite. + 199. (For ) If all , Dedekind finite subsets of are finite, then all Dedekind finite subsets of are finite. @@ -2049,7 +2049,7 @@ using the button in the bottom of the page. name="200"> - 200. For all infinite , . + 200. For all infinite , . @@ -2059,7 +2059,7 @@ using the button in the bottom of the page. name="201"> - 201. Linking Axiom for Boolean Algebras: Every Boolean algebra has a maximal linked system. ( is linked if for all and .) + 201. Linking Axiom for Boolean Algebras: Every Boolean algebra has a maximal linked system. ( is linked if for all and .) @@ -2069,7 +2069,7 @@ using the button in the bottom of the page. name="202"> - 202. : Every linearly ordered family of non-empty sets has a choice function. + 202. : Every linearly ordered family of non-empty sets has a choice function. @@ -2079,7 +2079,7 @@ using the button in the bottom of the page. name="203"> - 203. (disjoint,: Every partition of into non-empty subsets has a choice function. + 203. (disjoint,: Every partition of into non-empty subsets has a choice function. @@ -2089,7 +2089,7 @@ using the button in the bottom of the page. name="204"> - 204. For every infinite , there is a function from onto . + 204. For every infinite , there is a function from onto . @@ -2099,7 +2099,7 @@ using the button in the bottom of the page. name="205"> - 205. For all cardinals and , if and then there is a cardinal such that . + 205. For all cardinals and , if and then there is a cardinal such that . @@ -2109,7 +2109,7 @@ using the button in the bottom of the page. name="206"> - 206. The existence of a non-principal ultrafilter: There exists an infinite set and a non-principal ultrafilter on . + 206. The existence of a non-principal ultrafilter: There exists an infinite set and a non-principal ultrafilter on . @@ -2119,7 +2119,7 @@ using the button in the bottom of the page. name="207"> - 207. : The union of sets each of cardinality has cardinality less than . + 207. : The union of sets each of cardinality has cardinality less than . @@ -2129,7 +2129,7 @@ using the button in the bottom of the page. name="208"> - 208. For all ordinals , . + 208. For all ordinals , . @@ -2139,7 +2139,7 @@ using the button in the bottom of the page. name="209"> - 209. There is an ordinal such that for all , if is a denumerable union of denumerable sets then cannot be partitioned into non-empty sets. + 209. There is an ordinal such that for all , if is a denumerable union of denumerable sets then cannot be partitioned into non-empty sets. @@ -2159,7 +2159,7 @@ using the button in the bottom of the page. name="211"> - 211. : Dependent choice for relations on : If satisfies then there is a sequence of real numbers such that . + 211. : Dependent choice for relations on : If satisfies then there is a sequence of real numbers such that . @@ -2169,7 +2169,7 @@ using the button in the bottom of the page. name="212"> - 212. : If is a relation on such that for all , there is a such that , then there is a function such that for all , . + 212. : If is a relation on such that for all , there is a such that , then there is a function such that for all , . @@ -2179,7 +2179,7 @@ using the button in the bottom of the page. name="213"> - 213. : If then has a choice function. + 213. : If then has a choice function. @@ -2189,7 +2189,7 @@ using the button in the bottom of the page. name="214"> - 214. : For every family of infinite sets, there is a function such that for all , is a non-empty subset of and . + 214. : For every family of infinite sets, there is a function such that for all , is a non-empty subset of and . @@ -2199,7 +2199,7 @@ using the button in the bottom of the page. name="215"> - 215. If can be linearly ordered implies is finite), then is finite. + 215. If can be linearly ordered implies is finite), then is finite. @@ -2229,7 +2229,7 @@ using the button in the bottom of the page. name="218"> - 218. , relatively prime to ): , if is a set of non-empty sets, then there is a function such that for all , is a non-empty, finite subset of and is relatively prime to . + 218. , relatively prime to ): , if is a set of non-empty sets, then there is a function such that for all , is a non-empty, finite subset of and is relatively prime to . @@ -2239,7 +2239,7 @@ using the button in the bottom of the page. name="219"> - 219. , relatively prime to ): For all non-zero , if is a set of non-empty well orderable sets, then there is a function such that for all , is a non-empty, finite subset of , and is relatively prime to . + 219. , relatively prime to ): For all non-zero , if is a set of non-empty well orderable sets, then there is a function such that for all , is a non-empty, finite subset of , and is relatively prime to . @@ -2249,7 +2249,7 @@ using the button in the bottom of the page. name="220"> - 220. Suppose and is a prime. Any two elementary Abelian -groups (all non-trivial elements have order ) of the same cardinality are isomorphic. + 220. Suppose and is a prime. Any two elementary Abelian -groups (all non-trivial elements have order ) of the same cardinality are isomorphic. @@ -2259,7 +2259,7 @@ using the button in the bottom of the page. name="221"> - 221. For all infinite , there is a non-principal measure on . + 221. For all infinite , there is a non-principal measure on . @@ -2269,7 +2269,7 @@ using the button in the bottom of the page. name="222"> - 222. There is a non-principal measure on . + 222. There is a non-principal measure on . @@ -2279,7 +2279,7 @@ using the button in the bottom of the page. name="223"> - 223. There is an infinite set and a non-principal measure on . + 223. There is an infinite set and a non-principal measure on . @@ -2289,7 +2289,7 @@ using the button in the bottom of the page. name="224"> - 224. There is a partition of the real line into Borel sets such that for some , , . ( for is defined by induction, is an open subset of and for , if is even and if is odd.) + 224. There is a partition of the real line into Borel sets such that for some , , . ( for is defined by induction, is an open subset of and for , if is even and if is odd.) @@ -2299,7 +2299,7 @@ using the button in the bottom of the page. name="225"> - 225. Every proper filter on can be extended to an ultrafilter. + 225. Every proper filter on can be extended to an ultrafilter. @@ -2309,7 +2309,7 @@ using the button in the bottom of the page. name="226"> - 226. Let be a commutative ring with identity, a proper subring containing 1 and a prime ideal in . Then there is a subring of and a prime ideal in such that (a) (b) (c) is multiplicatively closed and (d) if , then is multiplicatively closed. + 226. Let be a commutative ring with identity, a proper subring containing 1 and a prime ideal in . Then there is a subring of and a prime ideal in such that (a) (b) (c) is multiplicatively closed and (d) if , then is multiplicatively closed. @@ -2319,7 +2319,7 @@ using the button in the bottom of the page. name="227"> - 227. For all groups , if every finite subgroup of can be fully ordered then can be fully ordered. + 227. For all groups , if every finite subgroup of can be fully ordered then can be fully ordered. @@ -2339,7 +2339,7 @@ using the button in the bottom of the page. name="229"> - 229. If is a partially ordered group, then can be extended to a linear order on if and only if for every finite set , with the identity for to , the signs () can be chosen so that (where is the normal sub-semi-group of generated by and where is the identity of .) + 229. If is a partially ordered group, then can be extended to a linear order on if and only if for every finite set , with the identity for to , the signs () can be chosen so that (where is the normal sub-semi-group of generated by and where is the identity of .) @@ -2349,7 +2349,7 @@ using the button in the bottom of the page. name="230"> - 230. . + 230. . @@ -2359,7 +2359,7 @@ using the button in the bottom of the page. name="231"> - 231. : The union of a well ordered collection of well orderable sets is well orderable. + 231. : The union of a well ordered collection of well orderable sets is well orderable. @@ -2369,7 +2369,7 @@ using the button in the bottom of the page. name="232"> - 232. Every metric space has a -point finite base. + 232. Every metric space has a -point finite base. @@ -2389,7 +2389,7 @@ using the button in the bottom of the page. name="234"> - 234. There is a non-Ramsey set: There is a set of infinite subsets of such that for every infinite subset of , has a subset which is in and a subset which is not in . + 234. There is a non-Ramsey set: There is a set of infinite subsets of such that for every infinite subset of , has a subset which is in and a subset which is not in . @@ -2399,7 +2399,7 @@ using the button in the bottom of the page. name="235"> - 235. If is a vector space and and are bases for then and are comparable. + 235. If is a vector space and and are bases for then and are comparable. @@ -2409,7 +2409,7 @@ using the button in the bottom of the page. name="236"> - 236. If is a vector space with a basis and is a linearly independent subset of such that no proper extension of is a basis for , then is a basis for . + 236. If is a vector space with a basis and is a linearly independent subset of such that no proper extension of is a basis for , then is a basis for . @@ -2419,7 +2419,7 @@ using the button in the bottom of the page. name="237"> - 237. The order of any group is divisible by the order of any of its subgroups, (i.e., if is a subgroup of then there is a set such that .) + 237. The order of any group is divisible by the order of any of its subgroups, (i.e., if is a subgroup of then there is a set such that .) @@ -2429,7 +2429,7 @@ using the button in the bottom of the page. name="238"> - 238. Every elementary Abelian group (that is, for some prime every non identity element has order ) is the direct sum of cyclic subgroups. + 238. Every elementary Abelian group (that is, for some prime every non identity element has order ) is the direct sum of cyclic subgroups. @@ -2439,7 +2439,7 @@ using the button in the bottom of the page. name="239"> - 239. AL20(): Every vector space over has the property that every linearly independent subset of can be extended to a basis. + 239. AL20(): Every vector space over has the property that every linearly independent subset of can be extended to a basis. @@ -2449,7 +2449,7 @@ using the button in the bottom of the page. name="240"> - 240. If a group satisfies every ascending chain of subgroups is finite," then every subgroup of is finitely generated. + 240. If a group satisfies every ascending chain of subgroups is finite," then every subgroup of is finitely generated. @@ -2459,7 +2459,7 @@ using the button in the bottom of the page. name="241"> - 241. Every algebraic closure of has a real closed subfield. + 241. Every algebraic closure of has a real closed subfield. @@ -2469,7 +2469,7 @@ using the button in the bottom of the page. name="242"> - 242. There is, up to an isomorphism, at most one algebraic closure of . + 242. There is, up to an isomorphism, at most one algebraic closure of . @@ -2499,7 +2499,7 @@ using the button in the bottom of the page. name="245"> - 245. There is a function such that for every , , is a function from onto . + 245. There is a function such that for every , , is a function from onto . @@ -2509,7 +2509,7 @@ using the button in the bottom of the page. name="246"> - 246. The monadic theory theory of is recursive. + 246. The monadic theory theory of is recursive. @@ -2529,7 +2529,7 @@ using the button in the bottom of the page. name="248"> - 248. For any , is the cardinal number of an infinite complete Boolean algebra if and only if . + 248. For any , is the cardinal number of an infinite complete Boolean algebra if and only if . @@ -2539,7 +2539,7 @@ using the button in the bottom of the page. name="249"> - 249. If is an infinite tree in which every element has exactly 2 immediate successors then has an infinite branch. + 249. If is an infinite tree in which every element has exactly 2 immediate successors then has an infinite branch. @@ -2549,7 +2549,7 @@ using the button in the bottom of the page. name="250"> - 250. : For every natural number , every well ordered family of element sets has a choice function. + 250. : For every natural number , every well ordered family of element sets has a choice function. @@ -2559,7 +2559,7 @@ using the button in the bottom of the page. name="251"> - 251. The additive groups and are isomorphic. + 251. The additive groups and are isomorphic. @@ -2569,7 +2569,7 @@ using the button in the bottom of the page. name="252"> - 252. The additive groups of and are isomorphic. + 252. The additive groups of and are isomorphic. @@ -2579,7 +2579,7 @@ using the button in the bottom of the page. name="253"> - 253. Łoś' Theorem: If is a relational system, any set and an ultrafilter in , then and are elementarily equivalent. + 253. Łoś' Theorem: If is a relational system, any set and an ultrafilter in , then and are elementarily equivalent. @@ -2589,7 +2589,7 @@ using the button in the bottom of the page. name="254"> - 254. : Every directed relation in which ramified subsets have least upper bounds, has a maximal element. + 254. : Every directed relation in which ramified subsets have least upper bounds, has a maximal element. @@ -2599,7 +2599,7 @@ using the button in the bottom of the page. name="255"> - 255. : Every directed relation in which every ramified subset has an upper bound, has a maximal element. + 255. : Every directed relation in which every ramified subset has an upper bound, has a maximal element. @@ -2609,7 +2609,7 @@ using the button in the bottom of the page. name="256"> - 256. : Every partially ordered set in which every forest has an upper bound, has a maximal element. + 256. : Every partially ordered set in which every forest has an upper bound, has a maximal element. @@ -2619,7 +2619,7 @@ using the button in the bottom of the page. name="257"> - 257. : Every transitive relation in which every partially ordered subset has an upper bound, has a maximal element. + 257. : Every transitive relation in which every partially ordered subset has an upper bound, has a maximal element. @@ -2629,7 +2629,7 @@ using the button in the bottom of the page. name="258"> - 258. : Every directed relation in which linearly ordered subsets have upper bounds, has a maximal element. + 258. : Every directed relation in which linearly ordered subsets have upper bounds, has a maximal element. @@ -2639,7 +2639,7 @@ using the button in the bottom of the page. name="259"> - 259. : If is a transitive and connected relation in which every well ordered subset has an upper bound, then has a maximal element. + 259. : If is a transitive and connected relation in which every well ordered subset has an upper bound, then has a maximal element. @@ -2649,7 +2649,7 @@ using the button in the bottom of the page. name="260"> - 260. : If is a transitive and connected relation in which every partially ordered subset has an upper bound, then has a maximal element. + 260. : If is a transitive and connected relation in which every partially ordered subset has an upper bound, then has a maximal element. @@ -2659,7 +2659,7 @@ using the button in the bottom of the page. name="261"> - 261. : Every transitive relation in which every subset which is a tree has an upper bound, has a maximal element. + 261. : Every transitive relation in which every subset which is a tree has an upper bound, has a maximal element. @@ -2669,7 +2669,7 @@ using the button in the bottom of the page. name="262"> - 262. : Every transitive relation in which every ramified subset has an upper bound, has a maximal element. + 262. : Every transitive relation in which every ramified subset has an upper bound, has a maximal element. @@ -2679,7 +2679,7 @@ using the button in the bottom of the page. name="263"> - 263. : Every every relation which is antisymmetric and connected contains a -maximal partially ordered subset. + 263. : Every every relation which is antisymmetric and connected contains a -maximal partially ordered subset. @@ -2689,7 +2689,7 @@ using the button in the bottom of the page. name="264"> - 264. : Every connected relation contains a -maximal partially ordered set. + 264. : Every connected relation contains a -maximal partially ordered set. @@ -2699,7 +2699,7 @@ using the button in the bottom of the page. name="265"> - 265. : Every relation contains a -maximal transitive subset. + 265. : Every relation contains a -maximal transitive subset. @@ -2709,7 +2709,7 @@ using the button in the bottom of the page. name="266"> - 266. : Every antisymmetric relation contains -maximal partially ordered subset. + 266. : Every antisymmetric relation contains -maximal partially ordered subset. @@ -2729,7 +2729,7 @@ using the button in the bottom of the page. name="268"> - 268. If is a lattice isomorphic to the lattice of subalgebras of some unary universal algebra (a unary universal algebra is one with only unary or nullary operations) and is an automorphism of of order 2 (that is, is the identity) then there is a unary algebra and an isomorphism from onto the lattice of subalgebras of with () for all . + 268. If is a lattice isomorphic to the lattice of subalgebras of some unary universal algebra (a unary universal algebra is one with only unary or nullary operations) and is an automorphism of of order 2 (that is, is the identity) then there is a unary algebra and an isomorphism from onto the lattice of subalgebras of with () for all . @@ -2739,7 +2739,7 @@ using the button in the bottom of the page. name="269"> - 269. For every cardinal , there is a set such that and there is a choice function on the collection of 2-element subsets of . + 269. For every cardinal , there is a set such that and there is a choice function on the collection of 2-element subsets of . @@ -2749,7 +2749,7 @@ using the button in the bottom of the page. name="270"> - 270. : The compactness theorem for propositional logic restricted to sets of formulas in which each variable occurs only in a finite number of formulas. + 270. : The compactness theorem for propositional logic restricted to sets of formulas in which each variable occurs only in a finite number of formulas. @@ -2759,7 +2759,7 @@ using the button in the bottom of the page. name="271"> - 271. If , : The compactness theorem for propositional logic restricted to sets of formulas in which each variable occurs in at most formulas. + 271. If , : The compactness theorem for propositional logic restricted to sets of formulas in which each variable occurs in at most formulas. @@ -2769,7 +2769,7 @@ using the button in the bottom of the page. name="272"> - 272. There is an such that neither nor has a perfect subset. + 272. There is an such that neither nor has a perfect subset. @@ -2789,7 +2789,7 @@ using the button in the bottom of the page. name="274"> - 274. There is a cardinal number and an such that adj. (The expression  adj means there are cardinals such that and and for all and if , then (Compare with [0 A]). + 274. There is a cardinal number and an such that adj. (The expression  adj means there are cardinals such that and and for all and if , then (Compare with [0 A]). @@ -2799,7 +2799,7 @@ using the button in the bottom of the page. name="275"> - 275. The sequence of cardinals has a unique minimal upper bound. + 275. The sequence of cardinals has a unique minimal upper bound. @@ -2809,7 +2809,7 @@ using the button in the bottom of the page. name="276"> - 276. : For every set , is Dedekind finite if and only if or . + 276. : For every set , is Dedekind finite if and only if or . @@ -2819,7 +2819,7 @@ using the button in the bottom of the page. name="277"> - 277. : Every non-well-orderable cardinal is decomposable. + 277. : Every non-well-orderable cardinal is decomposable. @@ -2829,7 +2829,7 @@ using the button in the bottom of the page. name="278"> - 278. In an integral domain , if every ideal is finitely generated then has a maximal proper ideal. + 278. In an integral domain , if every ideal is finitely generated then has a maximal proper ideal. @@ -2839,7 +2839,7 @@ using the button in the bottom of the page. name="279"> - 279. The Closed Graph Theorem for operations between Fréchet Spaces: Suppose and are Fréchet spaces, is linear and is closed in . Then is continuous. + 279. The Closed Graph Theorem for operations between Fréchet Spaces: Suppose and are Fréchet spaces, is linear and is closed in . Then is continuous. @@ -2859,7 +2859,7 @@ using the button in the bottom of the page. name="281"> - 281. There is a Hilbert space and an unbounded linear operator on . + 281. There is a Hilbert space and an unbounded linear operator on . @@ -2869,7 +2869,7 @@ using the button in the bottom of the page. name="282"> - 282. . + 282. . @@ -2879,7 +2879,7 @@ using the button in the bottom of the page. name="283"> - 283. Cardinality of well ordered subsets: For all and for all infinite , where is the set of all well orderable subsets of . + 283. Cardinality of well ordered subsets: For all and for all infinite , where is the set of all well orderable subsets of . @@ -2889,7 +2889,7 @@ using the button in the bottom of the page. name="284"> - 284. A system of linear equations over a field has a solution in if and only if every finite sub-system has a solution in . + 284. A system of linear equations over a field has a solution in if and only if every finite sub-system has a solution in . @@ -2899,7 +2899,7 @@ using the button in the bottom of the page. name="285"> - 285. Let be a set and , then has a fixed point if and only if is not the union of three mutually disjoint sets , and such that for . + 285. Let be a set and , then has a fixed point if and only if is not the union of three mutually disjoint sets , and such that for . @@ -2919,7 +2919,7 @@ using the button in the bottom of the page. name="287"> - 287. The Hahn-Banach Theorem for Separable Normed Linear Spaces: Assume is a separable normed linear space and satisfies and and assume is a linear function from a subspace of into which satisfies , then can be extended to so that is linear and . + 287. The Hahn-Banach Theorem for Separable Normed Linear Spaces: Assume is a separable normed linear space and satisfies and and assume is a linear function from a subspace of into which satisfies , then can be extended to so that is linear and . @@ -2929,7 +2929,7 @@ using the button in the bottom of the page. name="288"> - 288. If , : Every denumerable set of -element sets has a choice function. + 288. If , : Every denumerable set of -element sets has a choice function. @@ -2939,7 +2939,7 @@ using the button in the bottom of the page. name="289"> - 289. If is a set of subsets of a countable set and is closed under chain unions, then has a -maximal element. + 289. If is a set of subsets of a countable set and is closed under chain unions, then has a -maximal element. @@ -2949,7 +2949,7 @@ using the button in the bottom of the page. name="290"> - 290. For all infinite , . + 290. For all infinite , . @@ -2959,7 +2959,7 @@ using the button in the bottom of the page. name="291"> - 291. For all infinite , . + 291. For all infinite , . @@ -2969,7 +2969,7 @@ using the button in the bottom of the page. name="292"> - 292. : For each linearly ordered family of non-empty sets , there is a function such that for all is non-empty, finite subset of . + 292. : For each linearly ordered family of non-empty sets , there is a function such that for all is non-empty, finite subset of . @@ -2979,7 +2979,7 @@ using the button in the bottom of the page. name="293"> - 293. For all sets and , if can be linearly ordered and there is a mapping of onto , then can be linearly ordered. + 293. For all sets and , if can be linearly ordered and there is a mapping of onto , then can be linearly ordered. @@ -2989,7 +2989,7 @@ using the button in the bottom of the page. name="294"> - 294. Every linearly ordered -set is well orderable. + 294. Every linearly ordered -set is well orderable. @@ -3009,7 +3009,7 @@ using the button in the bottom of the page. name="296"> - 296. Part-: Every infinite set is the disjoint union of infinitely many infinite sets. + 296. Part-: Every infinite set is the disjoint union of infinitely many infinite sets. @@ -3079,7 +3079,7 @@ using the button in the bottom of the page. name="303"> - 303. If is a Boolean algebra, and is closed under , then there is a -maximal proper ideal of such that . + 303. If is a Boolean algebra, and is closed under , then there is a -maximal proper ideal of such that . @@ -3089,7 +3089,7 @@ using the button in the bottom of the page. name="304"> - 304. There does not exist a topological space such that every infinite subset of contains an infinite compact subset. + 304. There does not exist a topological space such that every infinite subset of contains an infinite compact subset. @@ -3099,7 +3099,7 @@ using the button in the bottom of the page. name="305"> - 305. There are Vitali equivalence classes. ( Vitali equivalence classes are equivalence classes of the real numbers under the relation .). + 305. There are Vitali equivalence classes. ( Vitali equivalence classes are equivalence classes of the real numbers under the relation .). @@ -3109,7 +3109,7 @@ using the button in the bottom of the page. name="306"> - 306. The set of Vitali equivalence classes is linearly orderable. ( Vitali equivalence classes are equivalence classes of the real numbers under the relation .). + 306. The set of Vitali equivalence classes is linearly orderable. ( Vitali equivalence classes are equivalence classes of the real numbers under the relation .). @@ -3119,7 +3119,7 @@ using the button in the bottom of the page. name="307"> - 307. If is the cardinality of the set of Vitali equivalence classes, then , where is Hartogs aleph function and the Vitali equivalence classes are equivalence classes of the real numbers under the relation . + 307. If is the cardinality of the set of Vitali equivalence classes, then , where is Hartogs aleph function and the Vitali equivalence classes are equivalence classes of the real numbers under the relation . @@ -3129,7 +3129,7 @@ using the button in the bottom of the page. name="308"> - 308. If is a prime and if is a set of finite groups, then the weak direct product has a maximal -subgroup. + 308. If is a prime and if is a set of finite groups, then the weak direct product has a maximal -subgroup. @@ -3139,7 +3139,7 @@ using the button in the bottom of the page. name="309"> - 309. The Banach-Tarski Paradox: There are three finite partitions , , and of such that is congruent to for and is congruent to for . + 309. The Banach-Tarski Paradox: There are three finite partitions , , and of such that is congruent to for and is congruent to for . @@ -3149,7 +3149,7 @@ using the button in the bottom of the page. name="310"> - 310. The Measure Extension Theorem: Suppose that is a subring (that is, and ) of a Boolean algebra and is a measure on (that is, , for , and .) then there is a measure on that extends . + 310. The Measure Extension Theorem: Suppose that is a subring (that is, and ) of a Boolean algebra and is a measure on (that is, , for , and .) then there is a measure on that extends . @@ -3159,7 +3159,7 @@ using the button in the bottom of the page. name="311"> - 311. Abelian groups are amenable. ( is amenable if there is a finitely additive measure on such that and , .) + 311. Abelian groups are amenable. ( is amenable if there is a finitely additive measure on such that and , .) @@ -3169,7 +3169,7 @@ using the button in the bottom of the page. name="312"> - 312. A subgroup of an amenable group is amenable. ( is amenable if there is a finitely additive measure on such that and , .) + 312. A subgroup of an amenable group is amenable. ( is amenable if there is a finitely additive measure on such that and , .) @@ -3179,7 +3179,7 @@ using the button in the bottom of the page. name="313"> - 313. (the set of integers under addition) is amenable. ( is amenable if there is a finitely additive measure on such that and , .) + 313. (the set of integers under addition) is amenable. ( is amenable if there is a finitely additive measure on such that and , .) @@ -3189,7 +3189,7 @@ using the button in the bottom of the page. name="314"> - 314. For every set and every permutation on there are two reflections and on such that and for every if then and . (A reflection is a permutation such that is the identity.) + 314. For every set and every permutation on there are two reflections and on such that and for every if then and . (A reflection is a permutation such that is the identity.) @@ -3199,7 +3199,7 @@ using the button in the bottom of the page. name="315"> - 315. , where + 315. , where @@ -3209,7 +3209,7 @@ using the button in the bottom of the page. name="316"> - 316. If a linearly ordered set has the fixed point property then is complete. ( has the fixed point property if every function satisfying has a fixed point, and ( is {\it complete} if every subset of has a least upper bound.) + 316. If a linearly ordered set has the fixed point property then is complete. ( has the fixed point property if every function satisfying has a fixed point, and ( is {\it complete} if every subset of has a least upper bound.) @@ -3219,7 +3219,7 @@ using the button in the bottom of the page. name="317"> - 317. Weak Sikorski Theorem: If is a complete, well orderable Boolean algebra and is a homomorphism of the Boolean algebra into where is a subalgebra of the Boolean algebra , then can be extended to a homomorphism of into . + 317. Weak Sikorski Theorem: If is a complete, well orderable Boolean algebra and is a homomorphism of the Boolean algebra into where is a subalgebra of the Boolean algebra , then can be extended to a homomorphism of into . @@ -3229,7 +3229,7 @@ using the button in the bottom of the page. name="318"> - 318. is not measurable. + 318. is not measurable. @@ -3249,7 +3249,7 @@ using the button in the bottom of the page. name="320"> - 320. No successor cardinal, , is measurable. + 320. No successor cardinal, , is measurable. @@ -3259,7 +3259,7 @@ using the button in the bottom of the page. name="321"> - 321. There does not exist an ordinal such that is weakly compact and is measurable. + 321. There does not exist an ordinal such that is weakly compact and is measurable. @@ -3269,7 +3269,7 @@ using the button in the bottom of the page. name="322"> - 322. , The Kinna-Wagner Selection Principle for a well ordered family of sets: For every well ordered set there is a function such that for all , if then . + 322. , The Kinna-Wagner Selection Principle for a well ordered family of sets: For every well ordered set there is a function such that for all , if then . @@ -3279,7 +3279,7 @@ using the button in the bottom of the page. name="323"> - 323. , The Kinna-Wagner Selection Principle for a family of well orderable sets: For every set of well orderable sets there is a function such that for all , if then . + 323. , The Kinna-Wagner Selection Principle for a family of well orderable sets: For every set of well orderable sets there is a function such that for all , if then . @@ -3289,7 +3289,7 @@ using the button in the bottom of the page. name="324"> - 324. , The Kinna-Wagner Selection Principle for a well ordered family of well orderable sets: For every well ordered set of well orderable sets, there is a function such that for all , if then . + 324. , The Kinna-Wagner Selection Principle for a well ordered family of well orderable sets: For every well ordered set of well orderable sets, there is a function such that for all , if then . @@ -3299,7 +3299,7 @@ using the button in the bottom of the page. name="325"> - 325. Ramsey's Theorem II: , if A is an infinite set and the family of all element subsets of is partitioned into sets , then there is an infinite subset such that all element subsets of belong to the same . (Also, see form 17.) + 325. Ramsey's Theorem II: , if A is an infinite set and the family of all element subsets of is partitioned into sets , then there is an infinite subset such that all element subsets of belong to the same . (Also, see form 17.) @@ -3309,7 +3309,7 @@ using the button in the bottom of the page. name="326"> - 326. 2-SAT: Restricted Compactness Theorem for Propositional Logic III: If is a set of formulas in a propositional language such that every finite subset of is satisfiable and if every formula in is a disjunction of at most two literals, then is satisfiable. (A literal is a propositional variable or its negation.) + 326. 2-SAT: Restricted Compactness Theorem for Propositional Logic III: If is a set of formulas in a propositional language such that every finite subset of is satisfiable and if every formula in is a disjunction of at most two literals, then is satisfiable. (A literal is a propositional variable or its negation.) @@ -3319,7 +3319,7 @@ using the button in the bottom of the page. name="327"> - 327. , The Kinna-Wagner Selection Principle for a well ordered family of finite sets: For every well ordered set of finite sets there is a function such that for all , if then . + 327. , The Kinna-Wagner Selection Principle for a well ordered family of finite sets: For every well ordered set of finite sets there is a function such that for all , if then . @@ -3329,7 +3329,7 @@ using the button in the bottom of the page. name="328"> - 328. : For every well ordered set such that for all , , there is a function such that and for every , is a finite, non-empty subset of . + 328. : For every well ordered set such that for all , , there is a function such that and for every , is a finite, non-empty subset of . @@ -3339,7 +3339,7 @@ using the button in the bottom of the page. name="329"> - 329. : For every set of well orderable sets such that for all , , there is a function such that for every , is a finite, non-empty subset of . + 329. : For every set of well orderable sets such that for all , , there is a function such that for every , is a finite, non-empty subset of . @@ -3349,7 +3349,7 @@ using the button in the bottom of the page. name="330"> - 330. : For every well ordered set of well orderable sets such that for all , , there is a function such that for every , is a finite, non-empty subset of . + 330. : For every well ordered set of well orderable sets such that for all , , there is a function such that for every , is a finite, non-empty subset of . @@ -3359,7 +3359,7 @@ using the button in the bottom of the page. name="331"> - 331. If is a family of compact non-empty topological spaces then there is a family such that , is an irreducible closed subset of . + 331. If is a family of compact non-empty topological spaces then there is a family such that , is an irreducible closed subset of . @@ -3379,7 +3379,7 @@ using the button in the bottom of the page. name="333"> - 333. : For every set of sets such that for all , , there is a function such that for every , is a finite, non-empty subset of and is odd. + 333. : For every set of sets such that for all , , there is a function such that for every , is a finite, non-empty subset of and is odd. @@ -3389,7 +3389,7 @@ using the button in the bottom of the page. name="334"> - 334. : For every set of sets such that for all , , there is a function such that for every , is a finite, non-empty subset of and is even. + 334. : For every set of sets such that for all , , there is a function such that for every , is a finite, non-empty subset of and is even. @@ -3399,7 +3399,7 @@ using the button in the bottom of the page. name="335"> - 335. Every quotient group of an Abelian group each of whose elements has order has a set of representatives. + 335. Every quotient group of an Abelian group each of whose elements has order has a set of representatives. @@ -3409,7 +3409,7 @@ using the button in the bottom of the page. name="336"> - 336. (For , .) For every infinite set , there is an infinite such that the set of all -element subsets of has a choice function. + 336. (For , .) For every infinite set , there is an infinite such that the set of all -element subsets of has a choice function. @@ -3419,7 +3419,7 @@ using the button in the bottom of the page. name="337"> - 337. : If is a well ordered collection of non-empty sets and there is a function defined on such that for every , is a linear ordering of , then there is a choice function for . + 337. : If is a well ordered collection of non-empty sets and there is a function defined on such that for every , is a linear ordering of , then there is a choice function for . @@ -3429,7 +3429,7 @@ using the button in the bottom of the page. name="338"> - 338. : The union of a denumerable number of denumerable sets is well orderable. + 338. : The union of a denumerable number of denumerable sets is well orderable. @@ -3439,7 +3439,7 @@ using the button in the bottom of the page. name="339"> - 339. Martin's Axiom : Whenever is a non-empty, ccc quasi-order (ccc means every anti-chain is countable) and is a family of dense subsets of , then there is a generic filter in . + 339. Martin's Axiom : Whenever is a non-empty, ccc quasi-order (ccc means every anti-chain is countable) and is a family of dense subsets of , then there is a generic filter in . @@ -3469,7 +3469,7 @@ using the button in the bottom of the page. name="342"> - 342. (For , .) : Every infinite family of -element sets has an infinite subfamily with a choice function. + 342. (For , .) : Every infinite family of -element sets has an infinite subfamily with a choice function. @@ -3479,7 +3479,7 @@ using the button in the bottom of the page. name="343"> - 343. A product of non-empty, compact topological spaces is non-empty. + 343. A product of non-empty, compact topological spaces is non-empty. @@ -3489,7 +3489,7 @@ using the button in the bottom of the page. name="344"> - 344. If is a family of non-empty sets, then there is a family such that , is an ultrafilter on . + 344. If is a family of non-empty sets, then there is a family such that , is an ultrafilter on . @@ -3499,7 +3499,7 @@ using the button in the bottom of the page. name="345"> - 345. Rasiowa-Sikorski Axiom: If is a Boolean algebra, is a non-zero element of , and is a denumerable set of subsets of then there is a maximal filter of such that and for each , if and exists then . + 345. Rasiowa-Sikorski Axiom: If is a Boolean algebra, is a non-zero element of , and is a denumerable set of subsets of then there is a maximal filter of such that and for each , if and exists then . @@ -3509,7 +3509,7 @@ using the button in the bottom of the page. name="346"> - 346. If is a vector space without a finite basis then contains an infinite, well ordered, linearly independent subset. + 346. If is a vector space without a finite basis then contains an infinite, well ordered, linearly independent subset. @@ -3519,7 +3519,7 @@ using the button in the bottom of the page. name="347"> - 347. Idemmultiple Partition Principle: If is idemmultiple () and , then . + 347. Idemmultiple Partition Principle: If is idemmultiple () and , then . @@ -3529,7 +3529,7 @@ using the button in the bottom of the page. name="348"> - 348. If is a group and and both freely generate then . + 348. If is a group and and both freely generate then . @@ -3539,7 +3539,7 @@ using the button in the bottom of the page. name="349"> - 349. : For every set of non-empty denumerable sets there is a function such that for all , is a finite, non-empty subset of . + 349. : For every set of non-empty denumerable sets there is a function such that for all , is a finite, non-empty subset of . @@ -3549,7 +3549,7 @@ using the button in the bottom of the page. name="350"> - 350. : For every denumerable set of non-empty denumerable sets there is a function such that for all , is a finite, non-empty subset of . + 350. : For every denumerable set of non-empty denumerable sets there is a function such that for all , is a finite, non-empty subset of . @@ -3589,7 +3589,7 @@ using the button in the bottom of the page. name="354"> - 354. A countable product of separable spaces is separable. + 354. A countable product of separable spaces is separable. @@ -3599,7 +3599,7 @@ using the button in the bottom of the page. name="355"> - 355. , The Kinna-Wagner Selection Principle for a denumerable family of sets: For every denumerable set there is a function such that for all , if then . + 355. , The Kinna-Wagner Selection Principle for a denumerable family of sets: For every denumerable set there is a function such that for all , if then . @@ -3609,7 +3609,7 @@ using the button in the bottom of the page. name="356"> - 356. , The Kinna-Wagner Selection Principle for a family of denumerable sets: For every set of denumerable sets there is a function such that for all , if then . + 356. , The Kinna-Wagner Selection Principle for a family of denumerable sets: For every set of denumerable sets there is a function such that for all , if then . @@ -3619,7 +3619,7 @@ using the button in the bottom of the page. name="357"> - 357. , The Kinna-Wagner Selection Principle for a denumerable family of denumerable sets: For every denumerable set of denumerable sets there is a function such that for all , if then . + 357. , The Kinna-Wagner Selection Principle for a denumerable family of denumerable sets: For every denumerable set of denumerable sets there is a function such that for all , if then . @@ -3629,7 +3629,7 @@ using the button in the bottom of the page. name="358"> - 358. , The Kinna-Wagner Selection Principle for a denumerable family of finite sets: For every denumerable set of finite sets there is a function such that for all , if then . + 358. , The Kinna-Wagner Selection Principle for a denumerable family of finite sets: For every denumerable set of finite sets there is a function such that for all , if then . @@ -3639,7 +3639,7 @@ using the button in the bottom of the page. name="359"> - 359. If and are families of pairwise disjoint sets and for all , then . + 359. If and are families of pairwise disjoint sets and for all , then . @@ -3649,7 +3649,7 @@ using the button in the bottom of the page. name="361"> - 361. In , the union of a denumerable number of analytic sets is analytic. + 361. In , the union of a denumerable number of analytic sets is analytic. @@ -3659,7 +3659,7 @@ using the button in the bottom of the page. name="362"> - 362. In , every Borel set is analytic. + 362. In , every Borel set is analytic. @@ -3669,7 +3669,7 @@ using the button in the bottom of the page. name="363"> - 363. There are exactly Borel sets in . + 363. There are exactly Borel sets in . @@ -3679,7 +3679,7 @@ using the button in the bottom of the page. name="364"> - 364. In , there is a measurable set that is not Borel. + 364. In , there is a measurable set that is not Borel. @@ -3689,7 +3689,7 @@ using the button in the bottom of the page. name="365"> - 365. For every uncountable set , if has the same cardinality as each of its uncountable subsets then . + 365. For every uncountable set , if has the same cardinality as each of its uncountable subsets then . @@ -3699,7 +3699,7 @@ using the button in the bottom of the page. name="366"> - 366. There is a discontinuous function such that for all real and , . + 366. There is a discontinuous function such that for all real and , . @@ -3709,7 +3709,7 @@ using the button in the bottom of the page. name="367"> - 367. There is a Hamel basis for as a vector space over . + 367. There is a Hamel basis for as a vector space over . @@ -3719,7 +3719,7 @@ using the button in the bottom of the page. name="368"> - 368. The set of all denumerable subsets of has power . + 368. The set of all denumerable subsets of has power . @@ -3729,7 +3729,7 @@ using the button in the bottom of the page. name="369"> - 369. If is partitioned into two sets, at least one of them has cardinality . + 369. If is partitioned into two sets, at least one of them has cardinality . @@ -3739,7 +3739,7 @@ using the button in the bottom of the page. name="370"> - 370. Weak Gelfand Extreme Point Theorem: If is a non-trivial Gelfand algebra then the closed unit ball in the dual of has an extreme point . + 370. Weak Gelfand Extreme Point Theorem: If is a non-trivial Gelfand algebra then the closed unit ball in the dual of has an extreme point . @@ -3759,7 +3759,7 @@ using the button in the bottom of the page. name="372"> - 372. Generalized Hahn-Banach Theorem: Assume that is a real vector space, is a Dedekind complete ordered vector space and is a subspace of . If is linear and is sublinear and if on then can be extended to a linear map such that on . + 372. Generalized Hahn-Banach Theorem: Assume that is a real vector space, is a Dedekind complete ordered vector space and is a subspace of . If is linear and is sublinear and if on then can be extended to a linear map such that on . @@ -3769,7 +3769,7 @@ using the button in the bottom of the page. name="373"> - 373. (For , .) : Every denumerable set of -element sets has an infinite subset with a choice function. + 373. (For , .) : Every denumerable set of -element sets has an infinite subset with a choice function. @@ -3779,7 +3779,7 @@ using the button in the bottom of the page. name="375"> - 375. Tietze-Urysohn Extension Theorem: If is a normal topological space, is closed in , and is continuous, then there exists a continuous function which extends . + 375. Tietze-Urysohn Extension Theorem: If is a normal topological space, is closed in , and is continuous, then there exists a continuous function which extends . @@ -3789,7 +3789,7 @@ using the button in the bottom of the page. name="376"> - 376. Restricted Kinna Wagner Principle: For every infinite set there is an infinite subset of and a function such that for every , if then is a non-empty proper subset of . De la Cruz/Di + 376. Restricted Kinna Wagner Principle: For every infinite set there is an infinite subset of and a function such that for every , if then is a non-empty proper subset of . De la Cruz/Di @@ -3799,7 +3799,7 @@ using the button in the bottom of the page. name="377"> - 377. Restricted Ordering Principle: For every infinite set there is an infinite subset of such that can be linearly ordered. De la Cruz/Di + 377. Restricted Ordering Principle: For every infinite set there is an infinite subset of such that can be linearly ordered. De la Cruz/Di @@ -3809,7 +3809,7 @@ using the button in the bottom of the page. name="378"> - 378. Restricted Choice for Families of Well Ordered Sets: For every infinite set there is an infinite subset of such that the family of non-empty well orderable subsets of has a choice function. De la Cruz/Di + 378. Restricted Choice for Families of Well Ordered Sets: For every infinite set there is an infinite subset of such that the family of non-empty well orderable subsets of has a choice function. De la Cruz/Di @@ -3819,7 +3819,7 @@ using the button in the bottom of the page. name="379"> - 379. : For every infinite family of sets each of which has at least two elements, there is an infinite subfamily of and a function such that for all , is a non-empty proper subset of . De la Cruz/Di + 379. : For every infinite family of sets each of which has at least two elements, there is an infinite subfamily of and a function such that for all , is a non-empty proper subset of . De la Cruz/Di @@ -3829,7 +3829,7 @@ using the button in the bottom of the page. name="380"> - 380. : For every infinite family of non-empty well orderable sets, there is an infinite subfamily of which has a choice function. De la Cruz/Di + 380. : For every infinite family of non-empty well orderable sets, there is an infinite subfamily of which has a choice function. De la Cruz/Di @@ -3859,7 +3859,7 @@ using the button in the bottom of the page. name="384"> - 384. Closed Filter Extendability for Spaces. Every closed filter in a topological space can be extended to a maximal closed filter. + 384. Closed Filter Extendability for Spaces. Every closed filter in a topological space can be extended to a maximal closed filter. @@ -3869,7 +3869,7 @@ using the button in the bottom of the page. name="385"> - 385. Countable Ultrafilter Theorem: Every proper filter with a countable base over a set (in ) can be extended to an ultrafilter. + 385. Countable Ultrafilter Theorem: Every proper filter with a countable base over a set (in ) can be extended to an ultrafilter. @@ -3889,7 +3889,7 @@ using the button in the bottom of the page. name="387"> - 387. DPO: Every infinite set has a non-trivial, dense partial order. (A partial ordering on a set is dense if and is non-trivial if ). + 387. DPO: Every infinite set has a non-trivial, dense partial order. (A partial ordering on a set is dense if and is non-trivial if ). @@ -3909,7 +3909,7 @@ using the button in the bottom of the page. name="389"> - 389. : Every denumerable family of two element subsets of has a choice function. + 389. : Every denumerable family of two element subsets of has a choice function. @@ -3929,7 +3929,7 @@ using the button in the bottom of the page. name="391"> - 391. : Every set of non-empty linearly orderable sets has a choice function. + 391. : Every set of non-empty linearly orderable sets has a choice function. @@ -3939,7 +3939,7 @@ using the button in the bottom of the page. name="392"> - 392. : Every linearly ordered set of linearly orderable sets has a choice function. + 392. : Every linearly ordered set of linearly orderable sets has a choice function. @@ -3949,7 +3949,7 @@ using the button in the bottom of the page. name="393"> - 393. : Every linearly ordered set of non-empty well orderable sets has a choice function. + 393. : Every linearly ordered set of non-empty well orderable sets has a choice function. @@ -3959,7 +3959,7 @@ using the button in the bottom of the page. name="394"> - 394. : Every well ordered set of non-empty linearly orderable sets has a choice function. + 394. : Every well ordered set of non-empty linearly orderable sets has a choice function. @@ -3969,7 +3969,7 @@ using the button in the bottom of the page. name="395"> - 395. : For each linearly ordered family of non-empty linearly orderable sets , there is a function such that for all is a non-empty, finite subset of . + 395. : For each linearly ordered family of non-empty linearly orderable sets , there is a function such that for all is a non-empty, finite subset of . @@ -3979,7 +3979,7 @@ using the button in the bottom of the page. name="396"> - 396. : For each linearly ordered family of non-empty well orderable sets , there is a function such that for all is a non-empty, finite subset of . + 396. : For each linearly ordered family of non-empty well orderable sets , there is a function such that for all is a non-empty, finite subset of . @@ -3989,7 +3989,7 @@ using the button in the bottom of the page. name="397"> - 397. : For each well ordered family of non-empty linearly orderable sets , there is a function such that for all is a non-empty, finite subset of . + 397. : For each well ordered family of non-empty linearly orderable sets , there is a function such that for all is a non-empty, finite subset of . @@ -3999,7 +3999,7 @@ using the button in the bottom of the page. name="398"> - 398. , The Kinna-Wagner Selection Principle for a linearly ordered family of sets: For every linearly ordered set there is a function such that for all , if then . + 398. , The Kinna-Wagner Selection Principle for a linearly ordered family of sets: For every linearly ordered set there is a function such that for all , if then . @@ -4009,7 +4009,7 @@ using the button in the bottom of the page. name="399"> - 399. , The Kinna-Wagner Selection Principle for a set of linearly orderable sets: For every set of linearly orderable sets there is a function such that for all , if then . + 399. , The Kinna-Wagner Selection Principle for a set of linearly orderable sets: For every set of linearly orderable sets there is a function such that for all , if then . @@ -4019,7 +4019,7 @@ using the button in the bottom of the page. name="400"> - 400. , The Kinna-Wagner Selection Principle for a linearly ordered set of linearly orderable sets: For every linearly ordered set of linearly orderable sets there is a function such that for all , if then . + 400. , The Kinna-Wagner Selection Principle for a linearly ordered set of linearly orderable sets: For every linearly ordered set of linearly orderable sets there is a function such that for all , if then . @@ -4029,7 +4029,7 @@ using the button in the bottom of the page. name="401"> - 401. , The Kinna-Wagner Selection Principle for a linearly ordered set of finite sets: For every linearly ordered set of finite sets there is a function such that for all , if then . + 401. , The Kinna-Wagner Selection Principle for a linearly ordered set of finite sets: For every linearly ordered set of finite sets there is a function such that for all , if then . @@ -4039,7 +4039,7 @@ using the button in the bottom of the page. name="402"> - 402. , The Kinna-Wagner Selection Principle for a well ordered set of linearly orderable sets: For every well ordered set of linearly orderable sets there is a function such that for all , if then . + 402. , The Kinna-Wagner Selection Principle for a well ordered set of linearly orderable sets: For every well ordered set of linearly orderable sets there is a function such that for all , if then . @@ -4049,7 +4049,7 @@ using the button in the bottom of the page. name="403"> - 403. , The Kinna-Wagner Selection Principle for a linearly ordered set of well orderable sets: For every linearly ordered set of well orderable sets there is a function such that for all , if then . + 403. , The Kinna-Wagner Selection Principle for a linearly ordered set of well orderable sets: For every linearly ordered set of well orderable sets there is a function such that for all , if then . @@ -4089,7 +4089,7 @@ using the button in the bottom of the page. name="407"> - 407. Let be a Boolean algebra, a non-zero element of and a sequence of subsets of such that for each , has a supremum . Then there exists an ultrafilter in such that and, for each , if , then . + 407. Let be a Boolean algebra, a non-zero element of and a sequence of subsets of such that for each , has a supremum . Then there exists an ultrafilter in such that and, for each , if , then . @@ -4099,7 +4099,7 @@ using the button in the bottom of the page. name="408"> - 408. If is a family of functions such that for each , , where and are non-empty sets, and is a filter base on such that 1. For all and all finite there is an such that is defined on , and \itemitem{2.} For all and all finite there exist at most finitely many functions on which are restrictions of the functions with , \noindent then there is a function with domain such that for each finite and each there is an such that . + 408. If is a family of functions such that for each , , where and are non-empty sets, and is a filter base on such that 1. For all and all finite there is an such that is defined on , and \itemitem{2.} For all and all finite there exist at most finitely many functions on which are restrictions of the functions with , \noindent then there is a function with domain such that for each finite and each there is an such that . @@ -4109,7 +4109,7 @@ using the button in the bottom of the page. name="409"> - 409. Suppose is a locally finite graph (i.e. is a non-empty set and is a function from to such that for each , and are finite), is a finite set of integers, and is a function mapping subsets of into subsets of . If for each finite subgraph there is a function such that for each , , then there is a function such that for all , . + 409. Suppose is a locally finite graph (i.e. is a non-empty set and is a function from to such that for each , and are finite), is a finite set of integers, and is a function mapping subsets of into subsets of . If for each finite subgraph there is a function such that for each , , then there is a function such that for all , . @@ -4149,7 +4149,7 @@ using the button in the bottom of the page. name="413"> - 413. Every infinite set is the union of a set, well-ordered by inclusion, of subsets which are non-equipollent to . + 413. Every infinite set is the union of a set, well-ordered by inclusion, of subsets which are non-equipollent to . @@ -4159,7 +4159,7 @@ using the button in the bottom of the page. name="414"> - 414. Every -frame is a -frame. + 414. Every -frame is a -frame. @@ -4169,7 +4169,7 @@ using the button in the bottom of the page. name="415"> - 415. Every -compactly generated complete lattice is algebraic. + 415. Every -compactly generated complete lattice is algebraic. @@ -4179,7 +4179,7 @@ using the button in the bottom of the page. name="416"> - 416. Every non-compact topological space is the union of a set that is well-ordered by inclusion and consists of open proper subsets of . + 416. Every non-compact topological space is the union of a set that is well-ordered by inclusion and consists of open proper subsets of . @@ -4199,7 +4199,7 @@ using the button in the bottom of the page. name="418"> - 418. DUM(): The countable disjoint union of metrizable spaces is metrizable. + 418. DUM(): The countable disjoint union of metrizable spaces is metrizable. @@ -4209,7 +4209,7 @@ using the button in the bottom of the page. name="419"> - 419. UT(,cuf,cuf): The union of a denumerable set of cuf sets is cuf. (A set is cuf if it is a countable union of finite sets.) + 419. UT(,cuf,cuf): The union of a denumerable set of cuf sets is cuf. (A set is cuf if it is a countable union of finite sets.) @@ -4219,7 +4219,7 @@ using the button in the bottom of the page. name="420"> - 420. UT(,,cuf): The union of a denumerable set of denumerable sets is cuf. + 420. UT(,,cuf): The union of a denumerable set of denumerable sets is cuf. @@ -4229,7 +4229,7 @@ using the button in the bottom of the page. name="421"> - 421. : The union of a denumerable set of well orderable sets can be well ordered. + 421. : The union of a denumerable set of well orderable sets can be well ordered. @@ -4239,7 +4239,7 @@ using the button in the bottom of the page. name="422"> - 422. , : The union of a well ordered set of element sets can be well ordered. + 422. , : The union of a well ordered set of element sets can be well ordered. @@ -4259,7 +4259,7 @@ using the button in the bottom of the page. name="425"> - 425. For every first countable topological space there is a family such that , countable local base at . + 425. For every first countable topological space there is a family such that , countable local base at . @@ -4269,7 +4269,7 @@ using the button in the bottom of the page. name="426"> - 426. If is a first countable topological space and is a family such that for all , is a local base at , then there is a family such that for every , is a countable local base at and . + 426. If is a first countable topological space and is a family such that for all , is a local base at , then there is a family such that for every , is a countable local base at and . @@ -4279,7 +4279,7 @@ using the button in the bottom of the page. name="427"> - 427. AL20(): There is a field such that every vector space over has the property that every independent subset of can be extended to a basis. + 427. AL20(): There is a field such that every vector space over has the property that every independent subset of can be extended to a basis. @@ -4289,7 +4289,7 @@ using the button in the bottom of the page. name="428"> - 428. B: There is a field such that every vector space over has a basis. + 428. B: There is a field such that every vector space over has a basis. @@ -4299,7 +4299,7 @@ using the button in the bottom of the page. name="429"> - 429. (Where is a prime) B: Every vector space over has a basis. ( is the element field.) + 429. (Where is a prime) B: Every vector space over has a basis. ( is the element field.) @@ -4309,7 +4309,7 @@ using the button in the bottom of the page. name="430"> - 430. (Where is a prime) AL21: Every vector space over has the property that for every subspace of , there is a subspace of such that and generates in other words such that . + 430. (Where is a prime) AL21: Every vector space over has the property that for every subspace of , there is a subspace of such that and generates in other words such that .