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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | isfin5-2 7901 | Alternate definition of V-finite which emphasizes the idempotent behavior of V-infinite sets. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
Fin^{V} | ||
Theorem | fin45 7902 | Every IV-finite set is V-finite: if we can pack two copies of the set into itself, we can certainly leave space. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Proof shortened by Mario Carneiro, 18-May-2015.) |
Fin^{IV} Fin^{V} | ||
Theorem | fin56 7903 | Every V-finite set is VI-finite because multiplication dominates addition for cardinals. (Contributed by Stefan O'Rear, 29-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
Fin^{V} Fin^{VI} | ||
Theorem | fin17 7904 | Every I-finite set is VII-finite. (Contributed by Mario Carneiro, 17-May-2015.) |
Fin^{VII} | ||
Theorem | fin67 7905 | Every VI-finite set is VII-finite. (Contributed by Stefan O'Rear, 29-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
Fin^{VI} Fin^{VII} | ||
Theorem | isfin7-2 7906 | A set is VII-finite iff it is non-well-orderable or finite. (Contributed by Mario Carneiro, 17-May-2015.) |
Fin^{VII} | ||
Theorem | fin71num 7907 | A well-orderable set is VII-finite iff it is I-finite. Thus even without choice, on the class of well-orderable sets all eight definitions of finite set coincide. (Contributed by Mario Carneiro, 18-May-2015.) |
Fin^{VII} | ||
Theorem | dffin7-2 7908 | Class form of isfin7-2 7906. (Contributed by Mario Carneiro, 17-May-2015.) |
Fin^{VII} | ||
Theorem | dfacfin7 7909 | Axiom of Choice equivalent: the VII-finite sets are the same as I-finite sets. (Contributed by Mario Carneiro, 18-May-2015.) |
CHOICE Fin^{VII} | ||
Theorem | fin1a2lem1 7910 | Lemma for fin1a2 7925. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
Theorem | fin1a2lem2 7911 | Lemma for fin1a2 7925. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
Theorem | fin1a2lem3 7912 | Lemma for fin1a2 7925. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
Theorem | fin1a2lem4 7913 | Lemma for fin1a2 7925. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
Theorem | fin1a2lem5 7914 | Lemma for fin1a2 7925. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
Theorem | fin1a2lem6 7915 | Lemma for fin1a2 7925. Establish that can be broken into two equipollent pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
Theorem | fin1a2lem7 7916* | Lemma for fin1a2 7925. Split a III-infinite set in two pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
Fin^{III} Fin^{III} Fin^{III} | ||
Theorem | fin1a2lem8 7917* | Lemma for fin1a2 7925. Split a III-infinite set in two pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
Fin^{III} Fin^{III} Fin^{III} | ||
Theorem | fin1a2lem9 7918* | Lemma for fin1a2 7925. In a chain of finite sets, initial segments are finite. (Contributed by Stefan O'Rear, 8-Nov-2014.) |
[] | ||
Theorem | fin1a2lem10 7919 | Lemma for fin1a2 7925. A nonempty finite union of members of a chain is a member of the chain. (Contributed by Stefan O'Rear, 8-Nov-2014.) |
[] | ||
Theorem | fin1a2lem11 7920* | Lemma for fin1a2 7925. (Contributed by Stefan O'Rear, 8-Nov-2014.) |
[] | ||
Theorem | fin1a2lem12 7921 | Lemma for fin1a2 7925. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
[] Fin^{III} | ||
Theorem | fin1a2lem13 7922 | Lemma for fin1a2 7925. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
[] Fin^{II} | ||
Theorem | fin12 7923 | Weak theorem which skips Ia but has a trivial proof, needed to prove fin1a2 7925. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
Fin^{II} | ||
Theorem | fin1a2s 7924* | An II-infinite set can have an I-infinite part broken off and remain II-infinite. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.) |
Fin^{II} Fin^{II} | ||
Theorem | fin1a2 7925 | Every Ia-finite set is II-finite. Theorem 1 of [Levy58], p. 3. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.) |
Fin^{Ia} Fin^{II} | ||
Theorem | itunifval 7926* | Function value of iterated unions. EDITORIAL: The iterated unions and order types of ordered sets are split out here because they could concievably be independently useful. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
Theorem | itunifn 7927* | Functionality of the iterated union. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
Theorem | ituni0 7928* | A zero-fold iterated union. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
Theorem | itunisuc 7929* | Successor iterated union. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
Theorem | itunitc1 7930* | Each union iterate is a member of the transitive closure. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
Theorem | itunitc 7931* | The union of all union iterates creates the transitive closure; compare trcl 7294. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
Theorem | ituniiun 7932* | Unwrap an iterated union from the "other end". (Contributed by Stefan O'Rear, 11-Feb-2015.) |
Theorem | hsmexlem7 7933* | Lemma for hsmex 7942. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
har har har | ||
Theorem | hsmexlem8 7934* | Lemma for hsmex 7942. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
har har har | ||
Theorem | hsmexlem9 7935* | Lemma for hsmex 7942. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
har har | ||
Theorem | hsmexlem1 7936 | Lemma for hsmex 7942. Bound the order type of a limited-cardinality set of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.) |
OrdIso ^{*} har | ||
Theorem | hsmexlem2 7937* | Lemma for hsmex 7942. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 8077 but use of order types allows to canonically choose the sub-bijections, removing the choice requirement. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.) |
OrdIso OrdIso har | ||
Theorem | hsmexlem3 7938* | Lemma for hsmex 7942. Clear hypothesis and extend previous result by dominance. Note that this could be substantially strengthened, e.g. using the weak Hartogs function, but all we need here is that there be *some* dominating ordinal. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.) |
OrdIso OrdIso ^{*} har | ||
Theorem | hsmexlem4 7939* | Lemma for hsmex 7942. The core induction, establishing bounds on the order types of iterated unions of the initial set. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
har har OrdIso | ||
Theorem | hsmexlem5 7940* | Lemma for hsmex 7942. Combining the above constraints, along with itunitc 7931 and tcrank 7438, gives an effective constraint on the rank of . (Contributed by Stefan O'Rear, 14-Feb-2015.) |
har har OrdIso har | ||
Theorem | hsmexlem6 7941* | Lemmr for hsmex 7942. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
har har OrdIso | ||
Theorem | hsmex 7942* | The collection of hereditarily size-limited well-founded sets comprise a set. The proof is that of Randall Holmes at http://math.boisestate.edu/~holmes/holmes/hereditary.pdf, with modifications to use Hartogs' theorem instead of the weak variant (inconsequentially weakening some intermediate results), and making the well-foundedness condition explicit to avoid a direct dependence on ax-reg 7190. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
Theorem | hsmex2 7943* | The set of hereditary size-limited sets, assuming ax-reg 7190. (Contributed by Stefan O'Rear, 11-Feb-2015.) |
Theorem | hsmex3 7944* | The set of hereditary size-limited sets, assuming ax-reg 7190, using strict comparison (an easy corrolary by separation). (Contributed by Stefan O'Rear, 11-Feb-2015.) |
In this section we add the Axiom of Choice ax-ac 7969, as well as weaker forms such as the axiom of countable choice ax-cc 7945 and dependent choice ax-dc 7956. We introduce these weaker forms so that theorems that do not need the full power of the axiom of choice, but need more than simple ZF, can use these intermediate axioms instead. The combination of the Zermel-Fraenkel axioms and the axiom of choice is often abbreviated as ZFC. The axiom of choice is widely accepted, and ZFC is the most commonly-accepted fundamental set of axioms for mathematics. However, there have been and still are some lingering controversies about the Axiom of Choice. The axiom of choice does not satisfy those who wish to have a constructive proof (e.g., it will not satify intuitionist logic). Thus, we make it easy to identify which proofs depend on the axiom of choice or its weaker forms. | ||
Axiom | ax-cc 7945* | The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 7988, but is weak enough that it can be proven using DC (see axcc 7968). It is, however, strictly stronger than ZF and cannot be proven in ZF. It states that any countable collection of non-empty sets must have a choice function. (Contributed by Mario Carneiro, 9-Feb-2013.) |
Theorem | axcc2lem 7946* | Lemma for axcc2 7947. (Contributed by Mario Carneiro, 8-Feb-2013.) |
Theorem | axcc2 7947* | A possibly more useful version of ax-cc using sequences instead of countable sets. The Axiom of Infinity is needed to prove this, and indeed this implies the Axiom of Infinity. (Contributed by Mario Carneiro, 8-Feb-2013.) |
Theorem | axcc3 7948* | A possibly more useful version of ax-cc 7945 using sequences instead of countable sets. The Axiom of Infinity is needed to prove this, and indeed this implies the Axiom of Infinity. (Contributed by Mario Carneiro, 8-Feb-2013.) (Revised by Mario Carneiro, 26-Dec-2014.) |
Theorem | axcc4 7949* | A version of axcc3 7948 that uses wffs instead of classes. (Contributed by Mario Carneiro, 7-Apr-2013.) |
Theorem | acncc 7950 | An ax-cc 7945 equivalent: every set has choice sets of length . (Contributed by Mario Carneiro, 31-Aug-2015.) |
AC | ||
Theorem | axcc4dom 7951* | Relax the constraint on axcc4 7949 to dominance instead of equinumerosity. (Contributed by Mario Carneiro, 18-Jan-2014.) |
Theorem | domtriomlem 7952* | Lemma for domtriom 7953. (Contributed by Mario Carneiro, 9-Feb-2013.) |
Theorem | domtriom 7953 | Trichotomy of equinumerosity for , proven using CC. Equivalently, all Dedekind-finite sets (as in isfin4-2 7824) are finite in the usual sense and conversely. (Contributed by Mario Carneiro, 9-Feb-2013.) |
Theorem | fin41 7954 | Under countable choice, the IV-finite sets (Dedekind-finite) coincide with I-finite (finite in the usual sense) sets. (Contributed by Mario Carneiro, 16-May-2015.) |
Fin^{IV} | ||
Theorem | dominf 7955 | A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 7945. See dominfac 8075 for a version proved from ax-ac 7969. The axiom of Regularity is used for this proof, via inf3lem6 7218, and its use is necessary: otherwise the set or (where the second example even has nonempty well-founded part) provides a counterexample. (Contributed by Mario Carneiro, 9-Feb-2013.) |
Axiom | ax-dc 7956* | Dependent Choice. Axiom DC1 of [Schechter] p. 149. This theorem is weaker than the Axiom of Choice but is stronger than Countable Choice. It shows the existence of a sequence whose values can only be shown to exist (but cannot be constructed explicitly) and also depend on earlier values in the sequence. Dependent choice is equivalent to the statement that every (nonempty) pruned tree has a branch. This axiom is redundant in ZFC; see axdc 8032. But ZF+DC is strictly weaker than ZF+AC, so this axiom provides for theorems that do not need the full power of AC. (Contributed by Mario Carneiro, 25-Jan-2013.) |
Theorem | dcomex 7957 | The Axiom of Dependent Choice implies Infinity, the way we have stated it. Thus we have Inf+AC implies DC and DC implies Inf, but AC does not imply Inf. (Contributed by Mario Carneiro, 25-Jan-2013.) |
Theorem | axdc2lem 7958* | Lemma for axdc2 7959. We construct a relation based on such that iff , and show that the "function" described by ax-dc 7956 can be restricted so that it is a real function (since the stated properties only show that it is the superset of a function). (Contributed by Mario Carneiro, 25-Jan-2013.) (Revised by Mario Carneiro, 26-Jun-2015.) |
Theorem | axdc2 7959* | An apparent strengthening of ax-dc 7956 (but derived from it) which shows that there is a denumerable sequence for any function that maps elements of a set to nonempty subsets of such that for all . The finitistic version of this can be proven by induction, but the infinite version requires this new axiom. (Contributed by Mario Carneiro, 25-Jan-2013.) |
Theorem | axdc3lem 7960* | The class of finite approximations to the DC sequence is a set. (We derive here the stronger statement that is a subset of a specific set, namely .) (Unnecessary distinct variable restrictions were removed by David Abernethy, 18-Mar-2014.) (Contributed by Mario Carneiro, 27-Jan-2013.) (Revised by Mario Carneiro, 18-Mar-2014.) |
Theorem | axdc3lem2 7961* | Lemma for axdc3 7964. We have constructed a "candidate set" , which consists of all finite sequences that satisfy our property of interest, namely on its domain, but with the added constraint that . These sets are possible "initial segments" of the infinite sequence satisfying these constraints, but we can leverage the standard ax-dc 7956 (with no initial condition) to select a sequence of ever-lengthening finite sequences, namely (for some integer ). We let our "choice" function select a sequence whose domain is one more than the last one, and agrees with the previous one on its domain. Thus, the application of vanilla ax-dc 7956 yields a sequence of sequences whose domains increase without bound, and whose union is a function which has all the properties we want. In this lemma, we show that given the sequence , we can construct the sequence that we are after. (Contributed by Mario Carneiro, 30-Jan-2013.) |
Theorem | axdc3lem3 7962* | Simple substitution lemma for axdc3 7964. (Contributed by Mario Carneiro, 27-Jan-2013.) |
Theorem | axdc3lem4 7963* | Lemma for axdc3 7964. We have constructed a "candidate set" , which consists of all finite sequences that satisfy our property of interest, namely on its domain, but with the added constraint that . These sets are possible "initial segments" of the infinite sequence satisfying these constraints, but we can leverage the standard ax-dc 7956 (with no initial condition) to select a sequence of ever-lengthening finite sequences, namely (for some integer ). We let our "choice" function select a sequence whose domain is one more than the last one, and agrees with the previous one on its domain. Thus, the application of vanilla ax-dc 7956 yields a sequence of sequences whose domains increase without bound, and whose union is a function which has all the properties we want. In this lemma, we show that is nonempty, and that always maps to a nonempty subset of , so that we can apply axdc2 7959. See axdc3lem2 7961 for the rest of the proof. (Contributed by Mario Carneiro, 27-Jan-2013.) |
Theorem | axdc3 7964* | Dependent Choice. Axiom DC1 of [Schechter] p. 149, with the addition of an initial value . This theorem is weaker than the Axiom of Choice but is stronger than Countable Choice. It shows the existence of a sequence whose values can only be shown to exist (but cannot be constructed explicitly) and also depend on earlier values in the sequence. (Contributed by Mario Carneiro, 27-Jan-2013.) |
Theorem | axdc4lem 7965* | Lemma for axdc4 7966. (Contributed by Mario Carneiro, 31-Jan-2013.) (Revised by Mario Carneiro, 16-Nov-2013.) |
Theorem | axdc4 7966* | A more general version of axdc3 7964 that allows the function to vary with . (Contributed by Mario Carneiro, 31-Jan-2013.) |
Theorem | axcclem 7967* | Lemma for axcc 7968. (Contributed by Mario Carneiro, 2-Feb-2013.) (Revised by Mario Carneiro, 16-Nov-2013.) |
Theorem | axcc 7968* | Although CC can be proven trivially using ac5 7988, we prove it here using DC. (Contributed by Mario Carneiro, 2-Feb-2013.) |
Axiom | ax-ac 7969* |
Axiom of Choice. The Axiom of Choice (AC) is usually considered an
extension of ZF set theory rather than a proper part of it. It is
sometimes considered philosophically controversial because it asserts
the existence of a set without telling us what the set is. ZF set
theory that includes AC is called ZFC.
The unpublished version given here says that given any set , there exists a that is a collection of unordered pairs, one pair for each non-empty member of . One entry in the pair is the member of , and the other entry is some arbitrary member of that member of . See the rewritten version ac3 7972 for a more detailed explanation. Theorem ac2 7971 shows an equivalent written compactly with restricted quantifiers. This version was specifically crafted to be short when expanded to primitives. Kurt Maes' 5-quantifier version ackm 7976 is slightly shorter when the biconditional of ax-ac 7969 is expanded into implication and negation. In axac3 7974 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 8175 (the Generalized Continuum Hypothesis implies the Axiom of Choice). Standard textbook versions of AC are derived as ac8 8003, ac5 7988, and ac7 7984. The Axiom of Regularity ax-reg 7190 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2 7641. Equivalents to AC are the well-ordering theorem weth 8006 and Zorn's lemma zorn 8018. See ac4 7986 for comments about stronger versions of AC. In order to avoid uses of ax-reg 7190 for derivation of AC equivalents, we provide ax-ac2 7973 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 7973 from ax-ac 7969 is shown by theorem axac2 7977, and the reverse derivation by axac 7978. (Contributed by NM, 18-Jul-1996.) |
Theorem | zfac 7970* | Axiom of Choice expressed with fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 7969. (Contributed by NM, 14-Aug-2003.) |
Theorem | ac2 7971* | Axiom of Choice equivalent. By using restricted quantifiers, we can express the Axiom of Choice with a single explicit conjunction. (If you want to figure it out, the rewritten equivalent ac3 7972 is easier to understand.) Note: aceq0 7629 shows the logical equivalence to ax-ac 7969. (Contributed by NM, 18-Jul-1996.) |
Theorem | ac3 7972* |
Axiom of Choice using abbreviations. The logical equivalence to ax-ac 7969
can be established by chaining aceq0 7629 and aceq2 7630. A standard
textbook version of AC is derived from this one in dfac2a 7640, and this
version of AC is derived from the textbook version in dfac2 7641.
The following sketch will help you understand this version of the axiom. Given any set , the axiom says that there exists a that is a collection of unordered pairs, one pair for each non-empty member of . One entry in the pair is the member of , and the other entry is some arbitrary member of that member of . Using the Axiom of Regularity, we can show that is really a set of ordered pairs, very similar to the ordered pair construction opthreg 7203. The key theorem for this (used in the proof of dfac2 7641) is preleq 7202. With this modified definition of ordered pair, it can be seen that is actually a choice function on the members of . For example, suppose . Let us try . For the member (of ) , the only assignment to and that satisfies the axiom is and , so there is exactly one as required. We verify the other two members of similarly. Thus satisfies the axiom. Using our modified ordered pair definition, we can say that corresponds to the choice function . Of course other choices for will also satisfy the axiom, for example . What AC tells us is that there exists at least one such , but it doesn't tell us which one. (Contributed by NM, 19-Jul-1996.) |
Axiom | ax-ac2 7973* | In order to avoid uses of ax-reg 7190 for derivation of AC equivalents, we provide ax-ac2 7973, which is equivalent to the standard AC of textbooks. This appears to be the shortest known equivalent to the standard AC when expressed in terms of set theory primitives. It was found by Kurt Maes as theorem ackm 7976. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1536 available. The derivation of ax-ac2 7973 from ax-ac 7969 is shown by theorem axac2 7977, and the reverse derivation by axac 7978. Note that we use ax-reg 7190 to derive ax-ac 7969 from ax-ac2 7973, but not to derive ax-ac2 7973 from ax-ac 7969. (Contributed by NM, 19-Dec-2016.) |
Theorem | axac3 7974 | This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 7973 as an axiom. (Contributed by Mario Carneiro, 6-May-2015.) (Revised by NM, 20-Dec-2016.) (Proof modification is discouraged.) |
CHOICE | ||
Theorem | axac3OLD 7975 | This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac 7969 as an axiom. Obsolete as of 20-Dec-2016. (Contributed by Mario Carneiro, 6-May-2015.) (New usage is discouraged.) |
CHOICE | ||
Theorem | ackm 7976* |
A remarkable equivalent to the Axiom of Choice that has only 5
quantifiers (when expanded to ,
primitives in prenex form),
discovered and proved by Kurt Maes. This establishes a new record,
reducing from 6 to 5 the largest number of quantified variables needed
by any ZFC axiom. The ZF-equivalence to AC is shown by theorem
dfackm 7676. Maes found this version of AC in April,
2004 (replacing a
longer version, also with 5 quantifiers, that he found in November,
2003). See Kurt Maes, "A 5-quantifier (,=)-expression
ZF-equivalent to the Axiom of Choice"
(http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.3162v1.pdf).
The original FOM posts are: http://www.cs.nyu.edu/pipermail/fom/2003-November/007631.html http://www.cs.nyu.edu/pipermail/fom/2003-November/007641.html. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.) (Proof modification is discouraged.) |
Theorem | axac2 7977* | Derive ax-ac2 7973 from ax-ac 7969. (Contributed by NM, 19-Dec-2016.) (Proof modification is discouraged.) |
Theorem | axac 7978* | Derive ax-ac 7969 from ax-ac2 7973. Note that ax-reg 7190 is used by the proof. (Contributed by NM, 19-Dec-2016.) (Proof modification is discouraged.) |
Theorem | axaci 7979 | Apply a choice equivalent. (Contributed by Mario Carneiro, 17-May-2015.) |
CHOICE | ||
Theorem | cardeqv 7980 | All sets are well-orderable under choice. (Contributed by Mario Carneiro, 28-Apr-2015.) |
Theorem | numth3 7981 | All sets are well-orderable under choice. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
Theorem | numth2 7982* | Numeration theorem: any set is equinumerous to some ordinal (using AC). Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 20-Oct-2003.) |
Theorem | numth 7983* | Numeration theorem: every set can be put into one-to-one correspondence with some ordinal (using AC). Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Mario Carneiro, 8-Jan-2015.) |
Theorem | ac7 7984* | An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 29-Apr-2004.) |
Theorem | ac7g 7985* | An Axiom of Choice equivalent similar to the Axiom of Choice (first form) of [Enderton] p. 49. (Contributed by NM, 23-Jul-2004.) |
Theorem | ac4 7986* |
Equivalent of Axiom of Choice. We do not insist that be a
function. However, theorem ac5 7988, derived from this one, shows that
this form of the axiom does imply that at least one such set whose
existence we assert is in fact a function. Axiom of Choice of
[TakeutiZaring] p. 83.
Takeuti and Zaring call this "weak choice" in contrast to "strong choice" , which asserts the existence of a universal choice function but requires second-order quantification on (proper) class variable and thus cannot be expressed in our first-order formalization. However, it has been shown that ZF plus strong choice is a conservative extension of ZF plus weak choice. See Ulrich Felgner, "Comparison of the axioms of local and universal choice," Fundamenta Mathematica, 71, 43-62 (1971). Weak choice can be strengthened in a different direction to choose from a collection of proper classes; see ac6s5 8002. (Contributed by NM, 21-Jul-1996.) |
Theorem | ac4c 7987* | Equivalent of Axiom of Choice (class version) (Contributed by NM, 10-Feb-1997.) |
Theorem | ac5 7988* | An Axiom of Choice equivalent: there exists a function (called a choice function) with domain that maps each nonempty member of the domain to an element of that member. Axiom AC of [BellMachover] p. 488. Note that the assertion that be a function is not necessary; see ac4 7986. (Contributed by NM, 29-Aug-1999.) |
Theorem | ac5b 7989* | Equivalent of Axiom of Choice. (Contributed by NM, 31-Aug-1999.) |
Theorem | ac6num 7990* | A version of ac6 7991 which takes the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.) |
Theorem | ac6 7991* | Equivalent of Axiom of Choice. This is useful for proving that there exists, for example, a sequence mapping natural numbers to members of a larger set , where depends on (the natural number) and (to specify a member of ). A stronger version of this theorem, ac6s 7995, allows to be a proper class. (Contributed by NM, 18-Oct-1999.) (Revised by Mario Carneiro, 27-Aug-2015.) |
Theorem | ac6c4 7992* | Equivalent of Axiom of Choice. is a collection of nonempty sets. (Contributed by Mario Carneiro, 22-Mar-2013.) |
Theorem | ac6c5 7993* | Equivalent of Axiom of Choice. is a collection of nonempty sets. Remark after Theorem 10.46 of [TakeutiZaring] p. 98. (Contributed by Mario Carneiro, 22-Mar-2013.) |
Theorem | ac9 7994* | An Axiom of Choice equivalent: the infinite Cartesian product of nonempty classes is nonempty. Axiom of Choice (second form) of [Enderton] p. 55 and its converse. (Contributed by Mario Carneiro, 22-Mar-2013.) |
Theorem | ac6s 7995* | Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 7447, we derive this strong version of ac6 7991 that doesn't require to be a set. (Contributed by NM, 4-Feb-2004.) |
Theorem | ac6n 7996* | Equivalent of Axiom of Choice. Contrapositive of ac6s 7995. (Contributed by NM, 10-Jun-2007.) |
Theorem | ac6s2 7997* | Generalization of the Axiom of Choice to classes. Slightly strengthened version of ac6s3 7998. (Contributed by NM, 29-Sep-2006.) |
Theorem | ac6s3 7998* | Generalization of the Axiom of Choice to classes. Theorem 10.46 of [TakeutiZaring] p. 97. (Contributed by NM, 3-Nov-2004.) |
Theorem | ac6sg 7999* | ac6s 7995 with sethood as antecedent. (Contributed by FL, 3-Aug-2009.) |
Theorem | ac6sf 8000* | Version of ac6 7991 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008.) |
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