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Theorem List for Metamath Proof Explorer - 8101-8200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremaxpowndlem3 8101* Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.) (Revised by Mario Carneiro, 10-Dec-2016.)

Theoremaxpowndlem4 8102 Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)

Theoremaxpownd 8103 A version of the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.)

Theoremaxregndlem1 8104 Lemma for the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)

Theoremaxregndlem2 8105* Lemma for the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)

Theoremaxregnd 8106 A version of the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)

Theoremaxinfndlem1 8107* Lemma for the Axiom of Infinity with no distinct variable conditions. (Contributed by NM, 5-Jan-2002.)

Theoremaxinfnd 8108 A version of the Axiom of Infinity with no distinct variable conditions. (Contributed by NM, 5-Jan-2002.)

Theoremaxacndlem1 8109 Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)

Theoremaxacndlem2 8110 Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)

Theoremaxacndlem3 8111 Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)

Theoremaxacndlem4 8112* Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 8-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)

Theoremaxacndlem5 8113* Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)

Theoremaxacnd 8114 A version of the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)

Theoremzfcndext 8115* Axiom of Extensionality ax-ext 2234, reproved from conditionless ZFC version and predicate calculus. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)

Theoremzfcndrep 8116* Axiom of Replacement ax-rep 4028, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)

Theoremzfcndun 8117* Axiom of Union ax-un 4403, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)

Theoremzfcndpow 8118* Axiom of Power Sets ax-pow 4082, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness," dtru 4095. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)

Theoremzfcndreg 8119* Axiom of Regularity ax-reg 7190, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)

Theoremzfcndinf 8120* Axiom of Infinity ax-inf 7223, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing theorem el 4086 in the proof. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)

Theoremzfcndac 8121* Axiom of Choice ax-ac 7969, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)

3.4  The Generalized Continuum Hypothesis

Syntaxcgch 8122 Extend class notation to include the collection of sets that satisfy the GCH.
GCH

Definitiondf-gch 8123* Define the collection of "GCH-sets", or sets for which the generalized continuum hypothesis holds. In this language the generalized continuum hypothesis can be expressed as GCH . A set satisfies the generalized continuum hypothesis if it is finite or there is no set strictly between and its powerset in cardinality. The continuum hypothesis is equivalent to GCH. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremelgch 8124* Elementhood in the collection of GCH-sets. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremfingch 8125 A finite set is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremgchi 8126 The only GCH-sets which have other sets between it and its power set are finite sets. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremgchen1 8127 If , and is an infinite GCH-set, then in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremgchen2 8128 If , and is an infinite GCH-set, then in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremgchor 8129 If , and is an infinite GCH-set, then either or in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremengch 8130 The property of being a GCH-set is a cardinal invariant. (Contributed by Mario Carneiro, 15-May-2015.)
GCH GCH

Theoremgchdomtri 8131 Under certain conditions, a GCH-set can demonstrate trichotomy of dominance. Lemma for gchac 8175. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremfpwwe2cbv 8132* Lemma for fpwwe2 8145. (Contributed by Mario Carneiro, 3-Jun-2015.)

Theoremfpwwe2lem1 8133* Lemma for fpwwe2 8145. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremfpwwe2lem2 8134* Lemma for fpwwe2 8145. (Contributed by Mario Carneiro, 19-May-2015.)

Theoremfpwwe2lem3 8135* Lemma for fpwwe2 8145. (Contributed by Mario Carneiro, 19-May-2015.)

Theoremfpwwe2lem5 8136* Lemma for fpwwe2 8145. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremfpwwe2lem6 8137* Lemma for fpwwe2 8145. (Contributed by Mario Carneiro, 18-May-2015.)
OrdIso        OrdIso

Theoremfpwwe2lem7 8138* Lemma for fpwwe2 8145. (Contributed by Mario Carneiro, 18-May-2015.)
OrdIso        OrdIso

Theoremfpwwe2lem8 8139* Lemma for fpwwe2 8145. Show by induction that the two isometries and agree on their common domain. (Contributed by Mario Carneiro, 15-May-2015.)
OrdIso        OrdIso

Theoremfpwwe2lem9 8140* Lemma for fpwwe2 8145. Given two well-orders and of parts of , one is an initial segment of the other. (The hypothesis is in order to break the symmetry of and .) (Contributed by Mario Carneiro, 15-May-2015.)
OrdIso        OrdIso

Theoremfpwwe2lem10 8141* Lemma for fpwwe2 8145. Given two well-orders and of parts of , one is an initial segment of the other. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremfpwwe2lem11 8142* Lemma for fpwwe2 8145. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremfpwwe2lem12 8143* Lemma for fpwwe2 8145. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremfpwwe2lem13 8144* Lemma for fpwwe2 8145. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremfpwwe2 8145* Given any function from well-orderings of subsets of to , there is a unique well-ordered subset which "agrees" with in the sense that each initial segment maps to its upper bound, and such that the entire set maps to an element of the set (so that it cannot be extended without losing the well-ordering). This theorem can be used to prove dfac8a 7541. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremfpwwecbv 8146* Lemma for fpwwe 8148. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremfpwwelem 8147* Lemma for fpwwe 8148. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremfpwwe 8148* Given any function from the powerset of to , canth2 6899 gives that the function is not injective, but we can say rather more than that. There is a unique well-ordered subset which "agrees" with in the sense that each initial segment maps to its upper bound, and such that the entire set maps to an element of the set (so that it cannot be extended without losing the well-ordering). This theorem can be used to prove dfac8a 7541. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremcanth4 8149* An "effective" form of Cantor's theorem canth 6178. For any function from the powerset of to , there are two definable sets and which witness non-injectivity of . Corollary 1.3 of [KanamoriPincus] p. 416. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremcanthnumlem 8150* Lemma for canthnum 8151. (Contributed by Mario Carneiro, 19-May-2015.)

Theoremcanthnum 8151 The set of well-orderable subsets of a set strictly dominates . A stronger form of canth2 6899. Corollary 1.4(a) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 19-May-2015.)

Theoremcanthwelem 8152* Lemma for canthnum 8151. (Contributed by Mario Carneiro, 31-May-2015.)

Theoremcanthwe 8153* The set of well-orders of a set strictly dominates . A stronger form of canth2 6899. Corollary 1.4(b) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 31-May-2015.)

Theoremcanthp1lem1 8154 Lemma for canthp1 8156. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremcanthp1lem2 8155* Lemma for canthp1 8156. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremcanthp1 8156 A slightly stronger form of Cantor's theorem: For , . Corollary 1.6 of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremfinngch 8157 The exclusion of finite sets from consideration in df-gch 8123 is necessary, because otherwise finite sets larger than a singleton would violate the GCH property. (Contributed by Mario Carneiro, 10-Jun-2015.)

Theoremgchcda1 8158 An infinite GCH-set is idempotent under cardinal successor. (Contributed by Mario Carneiro, 18-May-2015.)
GCH

Theoremgchinf 8159 An infinite GCH-set is Dedekind-infinite. (Contributed by Mario Carneiro, 31-May-2015.)
GCH

Theorempwfseqlem1 8160* Lemma for pwfseq 8166. Derive a contradiction by diagonalization. (Contributed by Mario Carneiro, 31-May-2015.)

Theorempwfseqlem2 8161* Lemma for pwfseq 8166. (Contributed by Mario Carneiro, 18-Nov-2014.)

Theorempwfseqlem3 8162* Lemma for pwfseq 8166. Using the construction from pwfseqlem1 8160, produce a function that maps any well-ordered infinite set to an element outside the set. (Contributed by Mario Carneiro, 31-May-2015.)

Theorempwfseqlem4a 8163* Lemma for pwfseqlem4 8164. (Contributed by Mario Carneiro, 7-Jun-2016.)

Theorempwfseqlem4 8164* Lemma for pwfseq 8166. Derive a final contradiction from the function in pwfseqlem3 8162. Applying fpwwe2 8145 to it, we get a certain maximal well-ordered subset , but the defining property contradicts our assumption on , so we are reduced to the case of finite. This too is a contradiction, though, because and its preimage under are distinct sets of the same cardinality and in a subset relation, which is impossible for finite sets. (Contributed by Mario Carneiro, 31-May-2015.)

Theorempwfseqlem5 8165* Lemma for pwfseq 8166. Although in some ways pwfseqlem4 8164 is the "main" part of the proof, one last aspect which makes up a remark in the original text is by far the hardest part to formalize. The main proof relies on the existence of an injection from the set of finite sequences on an infinite set to . Now this alone would not be difficult to prove; this is mostly the claim of fseqen 7538. However, what is needed for the proof is a canonical injection on these sets, so we have to start from scratch pulling together explicit bijections from the lemmas.

If one attempts such a program, it will mostly go through, but there is one key step which is inherently nonconstructive, namely the proof of infxpen 7526. The resolution is not obvious, but it turns out that reversing an infinite ordinal's Cantor normal form absorbs all the non-leading terms (cnfcom3c 7293), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 7529). (Contributed by Mario Carneiro, 31-May-2015.)

har        OrdIso                      seq𝜔

Theorempwfseq 8166* The powerset of a Dedekind-infinite set does not inject into the set of finite sequences. The proof is due to Halbeisen and Shelah. Proposition 1.7 of [KanamoriPincus] p. 418. (Contributed by Mario Carneiro, 31-May-2015.)

Theorempwxpndom2 8167 The powerset of a Dedekind-infinite set does not inject into its cross product with itself. (Contributed by Mario Carneiro, 31-May-2015.)

Theorempwxpndom 8168 The powerset of a Dedekind-infinite set does not inject into its cross product with itself. (Contributed by Mario Carneiro, 31-May-2015.)

Theorempwcdandom 8169 The powerset of a Dedekind-infinite set does not inject into its cardinal sum with itself. (Contributed by Mario Carneiro, 31-May-2015.)

Theoremgchcdaidm 8170 An infinite GCH-set is idempotent under cardinal sum. Part of Lemma 2.2 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 31-May-2015.)
GCH

Theoremgchxpidm 8171 An infinite GCH-set is idempotent under cardinal product. Part of Lemma 2.2 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 31-May-2015.)
GCH

Theoremgchaclem 8172 Lemma for gchac 8175 (obsolete, used in Sierpiński's proof). (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremgchhar 8173 A "local" form of gchac 8175. If and are GCH-sets, then the Hartogs number of is (so and a fortiori are well-orderable). The proof is due to Specker. Theorem 2.1 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 31-May-2015.)
GCH GCH har

Theoremgchacg 8174 A "local" form of gchac 8175. If and are GCH-sets, then is well-orderable. The proof is due to Specker. Theorem 2.1 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 15-May-2015.)
GCH GCH

Theoremgchac 8175 The Generalized Continuum Hypothesis implies the Axiom of Choice. The original proof is due to Sierpiński (1947); we use a refinement of Sierpiński's result due to Specker. (Contributed by Mario Carneiro, 15-May-2015.)
GCH CHOICE

Theoremgchpwdom 8176 A relationship between dominance over the powerset and strict dominance when the sets involved are infinite GCH-sets. Proposition 3.1 of [KanamoriPincus] p. 421. (Contributed by Mario Carneiro, 31-May-2015.)
GCH GCH

Theoremgchaleph 8177 If is a GCH-set and its powerset is well-orderable, then the successor aleph is equinumerous to the powerset of . (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremgchaleph2 8178 If and are GCH-sets, then the successor aleph is equinumerous to the powerset of . (Contributed by Mario Carneiro, 31-May-2015.)
GCH GCH

Theoremhargch 8179 If , then is a GCH-set. The much simpler converse to gchhar 8173. (Contributed by Mario Carneiro, 2-Jun-2015.)
har GCH

Theoremalephgch 8180 If is equinumerous to the powerset of , then is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremgch2 8181 It is sufficient to require that all alephs are GCH-sets to ensure the full generalized continuum hypothesis. (The proof uses the Axiom of Regularity.) (Contributed by Mario Carneiro, 15-May-2015.)
GCH GCH

Theoremgch3 8182 An equivalent formulation of the generalized continuum hypothesis. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremgch-kn 8183* The equivalence of two versions of the Generalized Continuum Hypothesis. The right-hand side is the standard version in the literature. The left-hand side is a version devised by Kannan Nambiar, which he calls the Axiom of Combinatorial Sets. For the notation and motivation behind this axiom, see his paper, "Derivation of Continuum Hypothesis from Axiom of Combinatorial Sets," available at http://www.e-atheneum.net/science/derivation_ch.pdf. The equivalence of the two sides provides a negative answer to Open Problem 2 in http://www.e-atheneum.net/science/open_problem_print.pdf. The key idea in the proof below is to equate both sides of alephexp2 8083 to the successor aleph using enen2 6887. (Contributed by NM, 1-Oct-2004.)

PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY

Here we introduce Tarski-Grothendieck (TG) set theory, named after mathematicians Alfred Tarski and Alexander Grothendieck. TG theory extends ZFC with the TG Axiom ax-groth 8325, which states that for every set there is an inaccessible cardinal such that is not in . The addition of this axiom to ZFC set theory provides a framework for category theory, thus for all practical purposes giving us a complete foundation for "all of mathematics."

We first introduce the concept of inaccessibles, including Weakly and strongly inaccessible cardinals (df-wina 8186 and df-ina 8187 respectively), Tarski's classes (df-tsk 8251), and a Grothendieck's universe (df-gru 8293). We then introduce the Tarski's axiom ax-groth 8325 and prove various properties from that.

4.1  Inaccessibles

4.1.1  Weakly and strongly inaccessible cardinals

Syntaxcwina 8184 The class of weak inaccessibles.

Syntaxcina 8185 The class of strong inaccessibles.

Definitiondf-wina 8186* An ordinal is weakly inaccessible iff it is a regular limit cardinal. Note that our definition allows as a weakly inacessible cardinal. (Contributed by Mario Carneiro, 22-Jun-2013.)

Definitiondf-ina 8187* An ordinal is strongly inaccessible iff it is a regular strong limit cardinal, which is to say that it dominates the powersets of every smaller ordinal. (Contributed by Mario Carneiro, 22-Jun-2013.)

Theoremelwina 8188* Conditions of weak inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.)

Theoremelina 8189* Conditions of strong inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.)

Theoremwinaon 8190 A weakly inaccessible cardinal is an ordinal. (Contributed by Mario Carneiro, 29-May-2014.)

Theoreminawinalem 8191* Lemma for inawina 8192. (Contributed by Mario Carneiro, 8-Jun-2014.)

Theoreminawina 8192 Every strongly inaccessible cardinal is weakly inaccessible. (Contributed by Mario Carneiro, 29-May-2014.)

Theoremomina 8193 is a strongly inaccessible cardinal. (Many definitions of "inaccessible" explicitly disallow as an inaccessible cardinal, but this choice allows us to reuse our results for inaccessibles for .) (Contributed by Mario Carneiro, 29-May-2014.)

Theoremwinacard 8194 A weakly inaccessible cardinal is a cardinal. (Contributed by Mario Carneiro, 29-May-2014.)

Theoremwinainflem 8195* A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.)

Theoremwinainf 8196 A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.)

Theoremwinalim 8197 A weakly inaccessible cardinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2014.)

Theoremwinalim2 8198* A nontrivial weakly inaccessible cardinal is a limit aleph. (Contributed by Mario Carneiro, 29-May-2014.)

Theoremwinafp 8199 A nontrivial weakly inaccessible cardinal is a fixed point of the aleph function. (Contributed by Mario Carneiro, 29-May-2014.)

Theoremwinafpi 8200 This theorem, which states that a nontrivial inaccessible cardinal is its own aleph number, is stated here in inference form, where the assumptions are in the hypotheses rather than an antecedent. Often, we use dedth 3511 to turn this type of statement into the closed form statement winafp 8199, but in this case, since it is consistent with ZFC that there are no nontrivial inaccessible cardinals, it is not possible to prove winafp 8199 using this theorem and dedth 3511, in ZFC. (You can prove this if you use ax-groth 8325, though.) (Contributed by Mario Carneiro, 28-May-2014.)

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