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Theorem List for Metamath Proof Explorer - 8501-8600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremgchen2 8501 If , and is an infinite GCH-set, then in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremgchor 8502 If , and is an infinite GCH-set, then either or in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremengch 8503 The property of being a GCH-set is a cardinal invariant. (Contributed by Mario Carneiro, 15-May-2015.)
GCH GCH

Theoremgchdomtri 8504 Under certain conditions, a GCH-set can demonstrate trichotomy of dominance. Lemma for gchac 8548. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremfpwwe2cbv 8505* Lemma for fpwwe2 8518. (Contributed by Mario Carneiro, 3-Jun-2015.)

Theoremfpwwe2lem1 8506* Lemma for fpwwe2 8518. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremfpwwe2lem2 8507* Lemma for fpwwe2 8518. (Contributed by Mario Carneiro, 19-May-2015.)

Theoremfpwwe2lem3 8508* Lemma for fpwwe2 8518. (Contributed by Mario Carneiro, 19-May-2015.)

Theoremfpwwe2lem5 8509* Lemma for fpwwe2 8518. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremfpwwe2lem6 8510* Lemma for fpwwe2 8518. (Contributed by Mario Carneiro, 18-May-2015.)
OrdIso        OrdIso

Theoremfpwwe2lem7 8511* Lemma for fpwwe2 8518. (Contributed by Mario Carneiro, 18-May-2015.)
OrdIso        OrdIso

Theoremfpwwe2lem8 8512* Lemma for fpwwe2 8518. Show by induction that the two isometries and agree on their common domain. (Contributed by Mario Carneiro, 15-May-2015.)
OrdIso        OrdIso

Theoremfpwwe2lem9 8513* Lemma for fpwwe2 8518. 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 8514* Lemma for fpwwe2 8518. Given two well-orders and of parts of , one is an initial segment of the other. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremfpwwe2lem11 8515* Lemma for fpwwe2 8518. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremfpwwe2lem12 8516* Lemma for fpwwe2 8518. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremfpwwe2lem13 8517* Lemma for fpwwe2 8518. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremfpwwe2 8518* 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 7911. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremfpwwecbv 8519* Lemma for fpwwe 8521. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremfpwwelem 8520* Lemma for fpwwe 8521. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremfpwwe 8521* Given any function from the powerset of to , canth2 7260 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 7911. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremcanth4 8522* An "effective" form of Cantor's theorem canth 6539. 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 8523* Lemma for canthnum 8524. (Contributed by Mario Carneiro, 19-May-2015.)

Theoremcanthnum 8524 The set of well-orderable subsets of a set strictly dominates . A stronger form of canth2 7260. Corollary 1.4(a) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 19-May-2015.)

Theoremcanthwelem 8525* Lemma for canthnum 8524. (Contributed by Mario Carneiro, 31-May-2015.)

Theoremcanthwe 8526* The set of well-orders of a set strictly dominates . A stronger form of canth2 7260. Corollary 1.4(b) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 31-May-2015.)

Theoremcanthp1lem1 8527 Lemma for canthp1 8529. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremcanthp1lem2 8528* Lemma for canthp1 8529. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremcanthp1 8529 A slightly stronger form of Cantor's theorem: For , . Corollary 1.6 of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 18-May-2015.)

Theoremfinngch 8530 The exclusion of finite sets from consideration in df-gch 8496 is necessary, because otherwise finite sets larger than a singleton would violate the GCH property. (Contributed by Mario Carneiro, 10-Jun-2015.)

Theoremgchcda1 8531 An infinite GCH-set is idempotent under cardinal successor. (Contributed by Mario Carneiro, 18-May-2015.)
GCH

Theoremgchinf 8532 An infinite GCH-set is Dedekind-infinite. (Contributed by Mario Carneiro, 31-May-2015.)
GCH

Theorempwfseqlem1 8533* Lemma for pwfseq 8539. Derive a contradiction by diagonalization. (Contributed by Mario Carneiro, 31-May-2015.)

Theorempwfseqlem2 8534* Lemma for pwfseq 8539. (Contributed by Mario Carneiro, 18-Nov-2014.)

Theorempwfseqlem3 8535* Lemma for pwfseq 8539. Using the construction from pwfseqlem1 8533, produce a function that maps any well-ordered infinite set to an element outside the set. (Contributed by Mario Carneiro, 31-May-2015.)

Theorempwfseqlem4a 8536* Lemma for pwfseqlem4 8537. (Contributed by Mario Carneiro, 7-Jun-2016.)

Theorempwfseqlem4 8537* Lemma for pwfseq 8539. Derive a final contradiction from the function in pwfseqlem3 8535. Applying fpwwe2 8518 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 8538* Lemma for pwfseq 8539. Although in some ways pwfseqlem4 8537 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 7908. 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 7896. 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 7663), which can be used to construct a pairing function explicitly using properties of the ordinal exponential (infxpenc 7899). (Contributed by Mario Carneiro, 31-May-2015.)

har        OrdIso                      seq𝜔

Theorempwfseq 8539* 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 8540 The powerset of a Dedekind-infinite set does not inject into its cross product with itself. (Contributed by Mario Carneiro, 31-May-2015.)

Theorempwxpndom 8541 The powerset of a Dedekind-infinite set does not inject into its cross product with itself. (Contributed by Mario Carneiro, 31-May-2015.)

Theorempwcdandom 8542 The powerset of a Dedekind-infinite set does not inject into its cardinal sum with itself. (Contributed by Mario Carneiro, 31-May-2015.)

Theoremgchcdaidm 8543 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 8544 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 8545 Lemma for gchac 8548 (obsolete, used in Sierpiński's proof). (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremgchhar 8546 A "local" form of gchac 8548. 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 8547 A "local" form of gchac 8548. 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 8548 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 8549 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 8550 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 8551 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 8552 If , then is a GCH-set. The much simpler converse to gchhar 8546. (Contributed by Mario Carneiro, 2-Jun-2015.)
har GCH

Theoremalephgch 8553 If is equinumerous to the powerset of , then is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremgch2 8554 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 8555 An equivalent formulation of the generalized continuum hypothesis. (Contributed by Mario Carneiro, 15-May-2015.)
GCH

Theoremgch-kn 8556* 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 8456 to the successor aleph using enen2 7248. (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 8698, 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 8559 and df-ina 8560 respectively), Tarski's classes (df-tsk 8624), and a Grothendieck's universe (df-gru 8666). We then introduce the Tarski's axiom ax-groth 8698 and prove various properties from that.

4.1  Inaccessibles

4.1.1  Weakly and strongly inaccessible cardinals

Syntaxcwina 8557 The class of weak inaccessibles.

Syntaxcina 8558 The class of strong inaccessibles.

Definitiondf-wina 8559* 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 8560* 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 8561* Conditions of weak inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.)

Theoremelina 8562* Conditions of strong inaccessibility. (Contributed by Mario Carneiro, 22-Jun-2013.)

Theoremwinaon 8563 A weakly inaccessible cardinal is an ordinal. (Contributed by Mario Carneiro, 29-May-2014.)

Theoreminawinalem 8564* Lemma for inawina 8565. (Contributed by Mario Carneiro, 8-Jun-2014.)

Theoreminawina 8565 Every strongly inaccessible cardinal is weakly inaccessible. (Contributed by Mario Carneiro, 29-May-2014.)

Theoremomina 8566 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 8567 A weakly inaccessible cardinal is a cardinal. (Contributed by Mario Carneiro, 29-May-2014.)

Theoremwinainflem 8568* A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.)

Theoremwinainf 8569 A weakly inaccessible cardinal is infinite. (Contributed by Mario Carneiro, 29-May-2014.)

Theoremwinalim 8570 A weakly inaccessible cardinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2014.)

Theoremwinalim2 8571* A nontrivial weakly inaccessible cardinal is a limit aleph. (Contributed by Mario Carneiro, 29-May-2014.)

Theoremwinafp 8572 A nontrivial weakly inaccessible cardinal is a fixed point of the aleph function. (Contributed by Mario Carneiro, 29-May-2014.)

Theoremwinafpi 8573 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 3780 to turn this type of statement into the closed form statement winafp 8572, but in this case, since it is consistent with ZFC that there are no nontrivial inaccessible cardinals, it is not possible to prove winafp 8572 using this theorem and dedth 3780, in ZFC. (You can prove this if you use ax-groth 8698, though.) (Contributed by Mario Carneiro, 28-May-2014.)

Theoremgchina 8574 Assuming the GCH, weakly and strongly inaccessible cardinals coincide. Theorem 11.20 of [TakeutiZaring] p. 106. (Contributed by Mario Carneiro, 5-Jun-2015.)
GCH

4.1.2  Weak universes

Syntaxcwun 8575 Extend class definition to include the class of all weak universes.
WUni

Syntaxcwunm 8576 Extend class definition to include the map whose value is the smallest weak universe.
wUniCl

Definitiondf-wun 8577* The class of all weak universes. A weak universe is a nonempty transitive class closed under union, pairing, and powerset. The advantage of a weak universe over a Grothendieck universe is that weak universes satisfy the analogue uniwun 8615 of grothtsk 8710 in ZFC. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Definitiondf-wunc 8578* A function that maps a set to the smallest weak universe that contains the elements of the set. (Contributed by Mario Carneiro, 2-Jan-2017.)
wUniCl WUni

Theoremiswun 8579* Properties of a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Theoremwuntr 8580 A weak universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Theoremwununi 8581 A weak universe is closed under union. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Theoremwunpw 8582 A weak universe is closed under powerset. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Theoremwunelss 8583 The elements of a weak universe are also subsets of it. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Theoremwunpr 8584 A weak universe is closed under pairing. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Theoremwunun 8585 A weak universe is closed under binary union. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Theoremwuntp 8586 A weak universe is closed under unordered triple. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Theoremwunss 8587 A weak universe is closed under subsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Theoremwunin 8588 A weak universe is closed under intersections. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Theoremwundif 8589 A weak universe is closed under set difference. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Theoremwunint 8590 A weak universe is closed under intersections. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Theoremwunsn 8591 A weak universe is closed under singletons. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Theoremwunsuc 8592 A weak universe is closed under successors. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Theoremwun0 8593 A weak universe contains the empty set. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Theoremwunr1om 8594 A weak universe is infinite, because it contains all the finite levels of the cumulative hierarchy. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Theoremwunom 8595 A weak universe contains all the finite ordinals, and hence is infinite. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Theoremwunfi 8596 A weak universe contains all finite sets with elements drawn from the universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Theoremwunop 8597 A weak universe is closed under ordered pairs. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Theoremwunot 8598 A weak universe is closed under ordered triples. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Theoremwunxp 8599 A weak universe is closed under cartesian products. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

Theoremwunpm 8600 A weak universe is closed under partial mappings. (Contributed by Mario Carneiro, 2-Jan-2017.)
WUni

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