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. ") (394 " 394.
: Every well ordered set of non-empty linearly orderable sets has a choice function. ") (250 " 250.
: For every natural number
, every well ordered family of
element sets has a choice function. ") (414 " 414. Every
-frame is a
-frame. ") (273 " 273. There is a subset of
which is not Borel.") (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
. ") (104 " 104. There is a regular uncountable aleph.") (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
. ") (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
. ") (121 " 121.
: Every linearly ordered set of non-empty finite sets has a choice function. ") (215 " 215. If
can be linearly ordered implies
is finite), then
is finite.") (241 " 241. Every algebraic closure of
has a real closed subfield. ") (221 " 221. For all infinite
, there is a non-principal measure on
. ") (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. ") (122 " 122.
: Every well ordered set of non-empty finite sets has a choice function. ") (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") (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") (300 " 300. Any continuous surjection between extremally disconnected compact Hausdorff spaces has an irreducible restriction to a closed subset of its domain. ") (110 " 110. Every vector space over
has a basis.") (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
,
.") (260 " 260.
: If
is a transitive and connected relation in which every partially ordered subset has an upper bound, then
has a maximal element.") (308 " 308
. If
is a prime and if
is a set of finite groups, then the weak direct product
has a maximal
-subgroup. ") (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") (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
. ") (255 " 255.
: Every directed relation
in which every ramified subset
has an upper bound, has a maximal element.") (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
. ") (382 " 382. DUMN: The disjoint union of metrizable spaces is normal.") (123 " 123.
: Uniform weak ultrafilter principle: For each family
of infinite sets
such that
,
is a non-principal ultrafilter on
.") (225 " 225. Every proper filter on
can be extended to an ultrafilter. ") (342 " 342
. (For
,
.)
: Every infinite family of
-element sets has an infinite subfamily with a choice function. ") (47 " 47
. If
,
: Every well ordered collection of
-element sets has a choice function.") (428 " 428.
B
: There is a field
such that every vector space over
has a basis.") (338 " 338.
: The union of a denumerable number of denumerable sets is well orderable. ") (336 " 336
. (For
,
.) For every infinite set
, there is an infinite
such that the set of all
-element subsets of
has a choice function. ") (265 " 265.
: Every relation
contains a
-maximal transitive subset. ") (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
. ") (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
.") (306 " 306. The set of Vitali equivalence classes is linearly orderable. ( Vitali equivalence classes are equivalence classes of the real numbers under the relation
.).") (32 " 32.
: Every denumerable set of non-empty countable sets has a choice function. ") (115 " 115. The product of weakly Loeb
spaces is weakly Loeb. ") (120 " 120
. If
,
: Every linearly ordered set of non-empty sets each of whose cardinality is in
has a choice function. ") (117 " 117. If
is a measurable cardinal, then
is the
th inaccessible cardinal.") (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
.") (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
.") (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
.") (124 " 124. Every operator on a Hilbert space with an amorphous base is the direct sum of a finite matrix and a scalar operator. (A set is amorphous if it is not the union of two disjoint infinite sets.) ") (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
.") (312 " 312. A subgroup of an amenable group is amenable. (
is amenable if there is a finitely additive measure
on
such that
and
,
.) ") (106 " 106. Baire Category Theorem for Compact Hausdorff Spaces: Every compact Hausdorff space is Baire.") (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
. ") (267 " 267. There is no infinite, free complete Boolean algebra. ") (62 " 62.
: Every set of non-empty finite sets has a choice function. ") (196 " 196
.
and
are not both measurable.") (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
.") (361 " 361. In
, the union of a denumerable number of analytic sets is analytic. ") (264 " 264.
: Every connected relation
contains a
-maximal partially ordered set.") (137 " 137
. Suppose
. If
is a 1-1 map from
into
then there are partitions
and
of
and
such that
maps
onto
. ") (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.") (37 " 37. Lebesgue measure is countably additive.") (51 " 51. Cofinality Principle: Every linear ordering has a cofinal sub well ordering. ") (173 " 173. MPL: Metric spaces are para-Lindelöf.") (48 " 48
. If
is a finite subset of
,
: For every
.") (341 " 341. Every Lindelöf metric space is second countable.") (34 " 34.
is regular.") (282 " 282.
.") (33 " 33
. If
,
: Every linearly ordered set of
element sets has a choice function.") (245 " 245. There is a function
such that for every
,
,
is a function from
onto
.") (353 " 353. A countable product of first countable spaces is first countable.") (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
.") (271 " 271
. If
,
: The compactness theorem for propositional logic restricted to sets of formulas in which each variable occurs in at most
formulas.") (90 " 90.
: Every linearly ordered set can be well ordered.") (58 " 58. There is an ordinal
such that
. (
is Hartogs' aleph, the least
not
.)") (82 " 82.
(see Howard/Yorke 1989): If
is infinite then
is Dedekind infinite. (
is finite
is Dedekind finite.)") (128 " 128. Aczel's Realization Principle: On every infinite set there is a Hausdorff topology with an infinite set of non-isolated points. ") (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.)") (152 " 152.
: Every non-well-orderable set is the union of a pairwise disjoint, well orderable family of denumerable sets. ") (189 " 189.
: For every Abelian group
there is an injective Abelian group
and a one to one homomorphism from
into
.") (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
. ") (277 " 277.
: Every non-well-orderable cardinal is decomposable.") (389 " 389.
: Every denumerable family of two element subsets of
has a choice function. ") (187 " 187. Every pair of cardinal numbers has a greatest lower bound (in the usual cardinal ordering.) ") (30 " 30. Ordering Principle: Every set can be linearly ordered. ") (222 " 222. There is a non-principal measure on
. ") (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
. ") (242 " 242. There is, up to an isomorphism, at most one algebraic closure of
. ") (294 " 294. Every linearly ordered
-set is well orderable. ") (111 " 111.
: The union of an infinite well ordered set of 2-element sets is an infinite well ordered set. ") (354 " 354. A countable product of separable
spaces is separable.") (346 " 346. If
is a vector space without a finite basis then
contains an infinite, well ordered, linearly independent subset.") (272 " 272. There is an
such that neither
nor
has a perfect subset.") (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
") (275 " 275. The sequence of cardinals
has a unique minimal upper bound. ") (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
. ") (385 " 385. Countable Ultrafilter Theorem: Every proper filter with a countable base over a set
(in
) can be extended to an ultrafilter.") (422 " 422
.
,
: The union of a well ordered set of
element sets can be well ordered. ") (344 " 344. If
is a family of non-empty sets, then there is a family
such that
,
is an ultrafilter on
. ") (80 " 80.
: Every denumerable set of pairs has a choice function. ") (89 " 89. Antichain Principle: Every partially ordered set has a maximal antichain.") (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.) ") (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.") (56 " 56.
. (
is Hartogs' aleph, the least
not
.) ") (420 " 420. UT(
,
,cuf): The union of a denumerable set of denumerable sets is cuf.") (4 " 4. Every infinite set is the union of some disjoint family of denumerable subsets. (Denumerable means
.)") (53 " 53. For all infinite cardinals
,
. ") (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
. ") (191 " 191.
: There is a set
such that for every set
, there is an ordinal
and a function from
onto
. ") (254 " 254.
: Every directed relation
in which ramified subsets have least upper bounds, has a maximal element.") (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.") (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
.) ") (25 " 25.
is regular for all ordinals
. ") (156 " 156. Theorem of Gelfand and Kolmogoroff: Two compact
spaces are homeomorphic if their rings of real valued continuous functions are isomorphic. ") (303 " 303. If
is a Boolean algebra,
and
is closed under
, then there is a
-maximal proper ideal
of
such that
. ") (288 " 288
. If
,
: Every denumerable set of
-element sets has a choice function.") (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
.)") (347 " 347. Idemmultiple Partition Principle: If
is idemmultiple (
) and
, then
.") (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
. ") (101 " 101. Partition Principle: If
is a partition of
, then
. ") (22 " 22.
: If every member of an infinite set of cardinality
has power
, then the union has power
.") (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. ") (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
. ") (404 " 404. Every infinite set can be partitioned into infinitely many sets, each of which has at least two elements.") (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.)") (413 " 413. Every infinite set
is the union of a set, well-ordered by inclusion, of subsets which are non-equipollent to
. ") (194 " 194.
or
: If
,
has domain
, and
is in
, then there is a sequence of elements
of
with
for all
.") (29 " 29. If
and
and
are families of pairwise disjoint sets and
for all
, then
. ") (172 " 172. For every infinite set
, if
is hereditarily countable (that is, every
is countable) then
.") (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
. ") (290 " 290. For all infinite
,
.") (109 " 109. Every field
and every vector space
over
has the property that each linearly independent set
can be extended to a basis.") (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.)") (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
. ") (281 " 281. There is a Hilbert space
and an unbounded linear operator on
. ") (348 " 348. If
is a group and
and
both freely generate
then
. ") (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
. ") (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. ") (41 " 41.
: For every cardinal
,
or
. ") (262 " 262.
: Every transitive relation
in which every ramified subset
has an upper bound, has a maximal element.") (203 " 203.
(disjoint,
: Every partition of
into non-empty subsets has a choice function.") (70 " 70. There is a non-trivial ultrafilter on
.") (164 " 164. Every non-well-orderable set has an infinite subset with a Dedekind finite power set.") (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
. ") (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") (73 " 73.
,
: For every
, if
is an infinite family of
element sets, then
has an infinite subfamily with a choice function. ") (103 " 103. If
is a linear ordering and
then some initial segment of
is uncountable.") (258 " 258.
: Every directed relation
in which linearly ordered subsets have upper bounds, has a maximal element.") (220 " 220
. Suppose
and
is a prime. Any two elementary Abelian
-groups (all non-trivial elements have order
) of the same cardinality are isomorphic.") (365 " 365. For every uncountable set
, if
has the same cardinality as each of its uncountable subsets then
. ") (134 " 134. If
is an infinite
space and
is
, then
is countable. (
is ``hereditarily
''.)") (411 " 411. RCuc (Reflexive Compactness for uniformly convex Banach spaces): The closed unit ball of a uniformly convex Banach space is compact for the weak topology.") (26 " 26.
: The union of denumerably many sets each of power
has power
. ") (246 " 246. The monadic theory theory
of
is recursive. ") (86 " 86
.
: If
is a set of non-empty sets such that
, then
has a choice function. ") (92 " 92.
: Every well ordered family of non-empty subsets of
has a choice function.") (368 " 368. The set of all denumerable subsets of
has power
. ") (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.) ") (60 " 60.
: Every set of non-empty, well orderable sets has a choice function.") (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.)") (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
. ") (20 " 20. If
and
are families of pairwise disjoint sets and
for all
, then
. ") (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
. ") (278 " 278. In an integral domain
, if every ideal is finitely generated then
has a maximal proper ideal.") (216 " 216. Every infinite tree has either an infinite chain or an infinite antichain. ") (7 " 7. There is no infinite decreasing sequence of cardinals.") (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
. ") (390 " 390. Every infinite set can be partitioned either into two infinite sets or infinitely many sets, each of which has at least two elements.") (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
.") (75 " 75. If a set has at least two elements, then it can be partitioned into well orderable subsets, each of which has at least two elements.") (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.") (153 " 153. The closed unit ball of a Hilbert space is compact in the weak topology.") (429 " 429
. (Where
is a prime) B: Every vector space over
has a basis. (
is the
element field.)") (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.") (359 " 359. If
and
are families of pairwise disjoint sets and
for all
, then
.") (163 " 163. Every non-well-orderable set has an infinite, Dedekind finite subset. ") (68 " 68. Nielsen-Schreier Theorem: Every subgroup of a free group is free. ") (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.") (136 " 136
. Surjective Cardinal Cancellation (depends on
): For all cardinals
and
,
implies
.") (46 " 46
. If
is a finite subset of
,
: For every
, every set of
-element sets has a choice function. ") (170 " 170.
.") (147 " 147.
: Every
topological space
can be covered by a well ordered family of discrete sets.") (10 " 10.
: Every denumerable family of non-empty finite sets has a choice function. ") (421 " 421.
: The union of a denumerable set of well orderable sets can be well ordered. ") (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
.") (179 " 179
. Suppose
is an ordinal.
,
).") (178 " 178
. If
,
and
,
,
: If
is any set of
-element sets then there is a function
with domain
such that for all
,
and
.") (369 " 369. If
is partitioned into two sets, at least one of them has cardinality
.") (309 " 309. The Banach-Tarski Paradox: There are three finite partitions
,
,
and
of
such that
is congruent to
for
and
is congruent to
for
. ") (235 " 235. If
is a vector space and
and
are bases for
then
and
are comparable.") (343 " 343. A product of non-empty, compact
topological spaces is non-empty. ") (98 " 98. The set of all finite subsets of a Dedekind finite set is Dedekind finite.") (116 " 116. Every compact
space is weakly Loeb. ") (35 " 35. The union of countably many meager subsets of
is meager. (Meager sets are the same as sets of the first category.)") (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
.") (230 " 230.
.") (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
). ") (8 " 8.
: Every denumerable family of non-empty sets has a choice function. ") (298 " 298. Every compact Hausdorff space has a Gleason cover.") (193 " 193.
: Every Abelian group is a homomorphic image of a free projective Abelian group. ") (6 " 6.
: The union of a denumerable family of denumerable subsets of
is denumerable. ") (393 " 393.
: Every linearly ordered set of non-empty well orderable sets has a choice function. ") (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
.)") (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
.") (301 " 301. Any continuous surjection between Boolean spaces has an irreducible restriction to a closed subset of its domain.") (181 " 181.
: Every set
of non-empty sets such that
has a choice function.") (313 " 313.
(the set of integers under addition) is amenable. (
is amenable if there is a finitely additive measure
on
such that
and
,
.) ") (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
. ") (243 " 243. Every principal ideal domain is a unique factorization domain. ") (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
. ") (315 " 315.
, where
") (418 " 418. DUM(
): The countable disjoint union of metrizable spaces is metrizable.") (213 " 213.
: If
then
has a choice function. ") (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
,
.") (305 " 305. There are
Vitali equivalence classes. ( Vitali equivalence classes are equivalence classes of the real numbers under the relation
.).") (36 " 36. Compact T
spaces are Loeb. (A space is Loeb if the set of non-empty closed sets has a choice function.)") (319 " 319. Measurable cardinals are inaccessible.") (162 " 162. Non-existence of infinite units: There is no infinite cardinal number
such that
and for all cardinals
and
,
or
.") (130 " 130.
is well orderable.") (259 " 259.
: If
is a transitive and connected relation in which every well ordered subset has an upper bound, then
has a maximal element. ") (340 " 340. Every Lindelöf metric space is separable.") (351 " 351. A countable product of metrizable spaces is metrizable.") (159 " 159. The regular cardinals are cofinal in the class of ordinals. ") (16 " 16.
: Every denumerable collection of non-empty sets each with power
has a choice function. ") (64 " 64.
(see Howard/Yorke 1989): There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.)") (276 " 276.
: For every set
,
is Dedekind finite if and only if
or
.") (177 " 177. An infinite box product of regular
spaces, each of cardinality greater than 1, is neither first countable nor connected. ") (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
. ") (144 " 144. Every set is almost well orderable.") (168 " 168. Dual Cantor-Bernstein Theorem:
and
implies
.") (286 " 286. Extended Krein-Milman Theorem: Let K be a quasicompact (sometimes called convex-compact), convex subset of a locally convex topological vector space, then K has an extreme point.") (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
. ") (263 " 263.
: Every every relation
which is antisymmetric and connected contains a
-maximal partially ordered subset. ") (118 " 118. Every linearly orderable topological space is normal. ") (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
,
. ") (251 " 251. The additive groups
and
are isomorphic. ") (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
. ") (85 " 85.
: Every family of denumerable sets has a choice function. ") (299 " 299. Any extremally disconnected compact Hausdorff space is projective in the category of Boolean topological spaces.") (256 " 256.
: Every partially ordered set
in which every forest
has an upper bound, has a maximal element.") (363 " 363. There are exactly
Borel sets in
. ") (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. ") (143 " 143.
: If
is a connected relation (
or
) then
contains a
-maximal transitive subset. ") (160 " 160. No Dedekind finite set can be mapped onto an aleph. ") (201 " 201. Linking Axiom for Boolean Algebras: Every Boolean algebra has a maximal linked system. (
is linked if
for all
and
.) ") (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
. ") (364 " 364. In
, there is a measurable set that is not Borel. ") (204 " 204. For every infinite
, there is a function from
onto
. ") (63 " 63.
: Weak ultrafilter principle: Every infinite set has a non-trivial ultrafilter.") (55 " 55. For all infinite cardinals
and
, if
then,
or
.") (252 " 252. The additive groups of
and
are isomorphic. ") (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. ") (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
.") (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.)") (2 " 2. Existence of successor cardinals: For every cardinal
there is a cardinal
such that
and
. ") (208 " 208. For all ordinals
,
. ") (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
.") (149 " 149.
: Every
topological space is a continuous, finite to one image of an A1 space.") (371 " 371. There is an infinite, compact, Hausdorff, extremally disconnected topological space. ") (266 " 266.
: Every antisymmetric relation contains
-maximal partially ordered subset.") (192 " 192.
sets: For every set
there is a projective set
and a function from
onto
. ") (54 " 54. For all infinite cardinals
,
adj
implies
is an aleph. (
adj
iff
and
.) ") (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 (
)
.") (232 " 232. Every metric space
has a
-point finite base. ") (79 " 79.
can be well ordered. ") (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
.") (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
.") (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. ") (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
. ") (211 " 211.
: Dependent choice for relations on
: If
satisfies
then there is a sequence
of real numbers such that
.") (302 " 302. Any continuous surjection between compact Hausdorff spaces has an irreducible restriction to a closed subset of its domain. ") (165 " 165.
: Every well ordered family of non-empty, well orderable sets has a choice function. ") (249 " 249. If
is an infinite tree in which every element has exactly 2 immediate successors then
has an infinite branch. ") (352 " 352. A countable product of second countable spaces is second countable.") (61 " 61.
)
: For each
,
, every set of
element sets has a choice function. ") (49 " 49. Order Extension Principle: Every partial ordering can be extended to a linear ordering. ") (227 " 227. For all groups
, if every finite subgroup of
can be fully ordered then
can be fully ordered.") (244 " 244. Every principal ideal domain has a maximal ideal.") (240 " 240. If a group
satisfies ``every ascending chain of subgroups is finite,\" then every subgroup of
is finitely generated.") (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
. ") (88 " 88.
: Every family of pairs has a choice function. ") (133 " 133. Every set is either well orderable or has an infinite amorphous subset. ") (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
. ") (257 " 257.
: Every transitive relation
in which every partially ordered subset has an upper bound, has a maximal element. ") (66 " 66. Every vector space over a field has a basis.") (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
.") (77 " 77. A linear ordering of a set
is a well ordering if and only if
has no infinite descending sequences.") (127 " 127. An amorphous power of a compact
space, which as a set is well orderable, is well orderable. ") (304 " 304. There does not exist a
topological space
such that every infinite subset of
contains an infinite compact subset.") (362 " 362. In
, every Borel set is analytic. ") (93 " 93. There is a non-measurable subset of
.") (69 " 69. Every field has an algebraic closure. ") (57 " 57. If
and
are Dedekind finite sets then either
or
. ") (412 " 412. RCh (Reflexive Compactness for Hilbert spaces): The closed unit ball of a Hilbert space is compact for the weak topology.") (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.)") (188 " 188.
: For every Abelian group
there is a projective Abelian group
and a homomorphism from
onto
.") (139 " 139. Using the discrete topology on 2,
is compact.") (83 " 83.
(see Howard/Yorke 1989):
-finite is equivalent to finite.") (195 " 195. Every general linear system has a linear global reaction. ") (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]).") (23 " 23.
: For every ordinal
, if
and every member of
has cardinality
, then
. ") (392 " 392.
: Every linearly ordered set of linearly orderable sets has a choice function. ") (335 " 335
. Every quotient group of an Abelian group each of whose elements has order
has a set of representatives.") (113 " 113. Tychonoff's Compactness Theorem for Countably Many Spaces: The product of a countable set of compact spaces is compact.") (206 " 206. The existence of a non-principal ultrafilter: There exists an infinite set
and a non-principal ultrafilter on
.") (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.") (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
.)") (311 " 311. Abelian groups are amenable. (
is amenable if there is a finitely additive measure
on
such that
and
,
.)") (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
.") (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
.") (405 " 405. Every infinite set can be partitioned into sets each of which is countable and has at least two elements. ") (1 " 1.
: The Axiom of Choice: Every set of non-empty sets has a choice function. ") (321 " 321. There does not exist an ordinal
such that
is weakly compact and
is measurable.") (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.)") (13 " 13. Every Dedekind finite subset of
is finite. ") (142 " 142.
: There is a set of reals without the property of Baire. ") (19 " 19. A real function is analytically representable if and only if it is in Baire's classification.") (388 " 388. Every infinite branching poset (a partially ordered set in which each element has at least two lower bounds) has either an infinite chain or an infinite antichain.") (200 " 200. For all infinite
,
.") (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
. ") (424 " 424. Every Lindelöf metric space is super second countable. ") (15 " 15.
(KW), The Kinna-Wagner Selection Principle: For every set
there is a function
such that for all
, if
then
. ") (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") (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)).") (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
.") (295 " 295. DO: Every infinite set has a dense linear ordering. ") (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
. ") (148 " 148.
: For every
topological space
, if
is well ordered, then
has a well ordered base.") (18 " 18.
: The union of a denumerable family of pairwise disjoint pairs has a denumerable subset.") (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
. ") (261 " 261.
: Every transitive relation
in which every subset which is a tree has an upper bound, has a maximal element.") (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.)") (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.)") (31 " 31.
: The countable union theorem: The union of a denumerable set of denumerable sets is denumerable. ") (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
. ") (205 " 205. For all cardinals
and
, if
and
then there is a cardinal
such that
.") (39 " 39.
: Every set
of non-empty sets such that
has a choice function.") (367 " 367. There is a Hamel basis for
as a vector space over
. ") (373 " 373
. (For
,
.)
: Every denumerable set of
-element sets has an infinite subset with a choice function. ") (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.") (185 " 185. Every linearly ordered Dedekind finite set is finite. ") (366 " 366. There is a discontinuous function
such that for all real
and
,
.") (24 " 24.
: Every denumerable collection of non-empty sets each with power
has a choice function. ") (289 " 289. If
is a set of subsets of a countable set and
is closed under chain unions, then
has a
-maximal element. ") (199 " 199
. (For
) If all
, Dedekind finite subsets of
are finite, then all
Dedekind finite subsets of
are finite. ") (190 " 190. There is a non-trivial injective Abelian group.") (154 " 154. Tychonoff's Compactness Theorem for Countably Many
Spaces: The product of countably many
compact spaces is compact.") (27 " 27.
: The union of denumerably many sets each of power
has power
. ") (210 " 210. The commutator subgroup of a free group is free.") (386 " 386. Every B compact (pseudo)metric space is Baire.") (214 " 214.
: For every family
of infinite sets, there is a function
such that for all
,
is a non-empty subset of
and
.") (40 " 40.
: Every well orderable set of non-empty sets has a choice function. ") (38 " 38.
is not the union of a countable family of countable sets. ") (415 " 415. Every
-compactly generated complete lattice is algebraic. ") (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
. ") (202 " 202.
: Every linearly ordered family of non-empty sets has a choice function. ") (384 " 384. Closed Filter Extendability for
Spaces. Every closed filter in a
topological space can be extended to a maximal closed filter.") (223 " 223. There is an infinite set
and a non-principal measure on
. ") (150 " 150.
: Every infinite set of denumerable sets has an infinite subset with a choice function. ") (238 " 238. Every elementary Abelian group (that is, for some prime
every non identity element has order
) is the direct sum of cyclic subgroups. ") (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
.) ") (231 " 231.
: The union of a well ordered collection of well orderable sets is well orderable.") (332 " 332. A product of non-empty compact sober topological spaces is non-empty. ") (102 " 102. For all Dedekind finite cardinals
and
, if
then
.") (21 " 21. If
is well ordered,
and
are families of pairwise disjoint sets, and
for all
, then
. ") (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
. ") (94 " 94.
: Every denumerable family of non-empty sets of reals has a choice function.") (158 " 158. In every Hilbert space
, if the closed unit ball is sequentially compact, then
has an orthonormal basis.") (318 " 318.
is not measurable.") (45 " 45
. If
,
: Every set of
-element sets has a choice function.") (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.)") (169 " 169. There is an uncountable subset of
without a perfect subset.") (132 " 132.
: Every infinite family of finite sets has an infinite subfamily with a choice function. ") (197 " 197.
is the union of three sets
with the property that for all
there is a straight line
such that
. ") (233 " 233. If a field has an algebraic closure it is unique up to isomorphism.") (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.)") (97 " 97. Cardinal Representatives: For every set
there is a function
with domain
such that for all
, (i)
and (ii)
. ") (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.) ") (14 " 14. BPI: Every Boolean algebra has a prime ideal.") (228 " 228. Every torsion free Abelian group can be fully ordered.") (293 " 293. For all sets
and
, if
can be linearly ordered and there is a mapping of
onto
, then
can be linearly ordered. ") (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
.) ") (96 " 96. Löwig's Theorem. If
and
are both bases for the vector space
then
.") (247 " 247. Every atomless Boolean algebra is Dedekind infinite. ") (67 " 67.
(MC), The Axiom of Multiple Choice: For every set
of non-empty sets there is a function
such that
and
is finite). ") (9 " 9. Finite
Dedekind finite:
(see Jech 1973b):
(\\ac{Howard/Yorke} \\cite{1989}): Every Dedekind finite set is finite.") (207 " 207
.
: The union of
sets each of cardinality
has cardinality less than
. ") (253 " 253. Łoś' Theorem: If
is a relational system,
any set and
an ultrafilter in
, then
and
are elementarily equivalent. ") (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
.) ") (71 " 71
.
:
or
.") (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
.") (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
. ") (297 " 297. Extremally disconnected compact Hausdorff spaces are projective in the category of all compact Hausdorff spaces.") (217 " 217. Every infinite partially ordered set has either an infinite chain or an infinite antichain.") (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
.") (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. ") (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.") (176 " 176. Every infinite, locally finite group has an infinite Abelian subgroup. ( Locally finite means every finite subset generates a finite subgroup.) ") (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
. ") (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") (248 " 248. For any
,
is the cardinal number of an infinite complete Boolean algebra if and only if
. ") (410 " 410. RC (Reflexive Compactness): The closed unit ball of a reflexive normed space is compact for the weak topology.") (3 " 3.
: For all infinite cardinals
,
.") (291 " 291. For all infinite
,
.") (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
. ") (5 " 5.
: Every denumerable set of non-empty denumerable subsets of
has a choice function.") (391 " 391.
: Every set of non-empty linearly orderable sets has a choice function. ") (381 " 381. DUM: The disjoint union of metrizable spaces is metrizable.") (320 " 320. No successor cardinal,
, is measurable.") (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
. ") (100 " 100. Weak Partition Principle: For all sets
and
, if
, then it is not the case that
. ") (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
. ") (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.") (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
. ") (280 " 280. There is a complete separable metric space with a subset which does not have the Baire property.") (186 " 186. Every pair of cardinal numbers has a least upper bound (in the usual cardinal ordering.) ") (425 " 425. For every first countable topological space
there is a family
such that
,
countable local base at
. ") (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
.") (296 " 296. Part-
: Every infinite set is the disjoint union of infinitely many infinite sets. ") (417 " 417. On every non-trivial Banach space there is a non-trivial linear functional (bounded or unbounded). ") (406 " 406. The product of compact Hausdorf spaces is countably compact.") (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
.") (84 " 84.
(see Howard/Yorke 1989):
is
-finite if and only if
is Dedekind finite). ") (108 " 108. There is an ordinal
such that
is not the union of a denumerable set of denumerable sets.") (151 " 151.
(
): The union of a well ordered set of denumerable sets is well orderable.") (239 " 239. AL20(
): Every vector
space over
has the property that every linearly independent subset of
can be extended to a basis.") (283 " 283. Cardinality of well ordered subsets: For all
and for all infinite
,
where
is the set of all well orderable subsets of
. ") (166 " 166.
: Every infinite family of pairs has an infinite subfamily with a choice function.") (91 " 91.
: The power set of a well ordered set can be well ordered. ") (182 " 182. There is an aleph whose cofinality is greater than
. "))
\ No newline at end of file