Movatterモバイル変換

Continuum hypothesis

From Wikipedia, the free encyclopedia

Proposition in mathematical logic

This article is about the hypothesis in set theory. For the assumption in fluid mechanics, seeContinuum assumption. For the album by Epoch of Unlight, seeThe Continuum Hypothesis (album).

Inmathematics, specificallyset theory, thecontinuum hypothesis (abbreviatedCH) is a hypothesis about the possible sizes ofinfinite sets. It states:

There is no set whosecardinality is strictly between that of theintegers and thereal numbers.

InZermelo–Fraenkel set theory with theaxiom of choice (ZFC), this is equivalent to the following equation inaleph numbers: $2^{\aleph _{0}}=\aleph _{1}$ , or even shorter withbeth numbers: $\beth _{1}=\aleph _{1}$ .

The continuum hypothesis was advanced byGeorg Cantor in 1878,^[1] and establishing its truth or falsehood is the first ofHilbert's 23 problems presented in 1900. The answer to this problem isindependent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 byPaul Cohen, complementing earlier work byKurt Gödel in 1940.^[2]

The name of the hypothesis comes from the termcontinuum for the real numbers.

History

[edit]

Cantor believed the continuum hypothesis to be true and for many years tried in vain to prove it.^[3] It became the first on David Hilbert'slist of important open questions that was presented at theInternational Congress of Mathematicians in the year 1900 in Paris.Axiomatic set theory was at that point not yet formulated.Kurt Gödel proved in 1940 that the negation of the continuum hypothesis, i.e., the existence of a set with intermediate cardinality, could not be proved in standard set theory.^[2] The second half of the independence of the continuum hypothesis – i.e., unprovability of the nonexistence of an intermediate-sized set – was proved in 1963 byPaul Cohen.^[4]

Cardinality of infinite sets

[edit]

Main article:Cardinal number

Two sets are said to have the samecardinality orcardinal number if there exists abijection (a one-to-one correspondence) between them. Intuitively, for two sets $S {\displaystyle S}$ and $T {\displaystyle T}$ to have the same cardinality means that it is possible to "pair off" elements of $S {\displaystyle S}$ with elements of $T {\displaystyle T}$ in such a fashion that every element of $S {\displaystyle S}$ is paired off with exactly one element of $T {\displaystyle T}$ and vice versa. Hence, the set $\{{\text{banana}},{\text{apple}},{\text{pear}}\}$ has the same cardinality as $\{{\text{yellow}},{\text{red}},{\text{green}}\}$ despite the sets themselves containing different elements.

With infinite sets such as the set ofintegers orrational numbers, the existence of a bijection between two sets becomes more difficult to demonstrate. The rational numbers $\mathbb {Q}$ seemingly form a counterexample to the continuum hypothesis: the integers form a proper subset of the rationals, which themselves form a proper subset of the reals, so intuitively, there are more rational numbers than integers and more real numbers than rational numbers. However, this intuitive analysis is flawed since it does not take into account the fact that all three sets areinfinite. Perhaps more importantly, it in fact conflates the concept of "size" of the set $\mathbb {Q}$ with the order or topological structure placed on it. In fact, it turns out the rational numbers can actually be placed in one-to-one correspondence with the integers, and therefore the set of rational numbers is the same size (cardinality) as the set of integers: they are bothcountable sets.^[5]

Cantor gave two proofs that the cardinality of the set ofintegers is strictly smaller than that of the set ofreal numbers (seeCantor's first uncountability proof andCantor's diagonal argument). His proofs, however, give no indication of the extent to which the cardinality of the integers is less than that of the real numbers. Cantor proposed the continuum hypothesis as a possible solution to this question.

In simple terms, the Continuum Hypothesis (CH) states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers. That is, every infinite set $S\subseteq \mathbb {R}$ of real numbers can either be mapped one-to-one into the integers or the real numbers can be mapped one-to-one into $S {\displaystyle S}$ . Since the real numbers areequinumerous with thepowerset of the integers, i.e. $|\mathbb {R} |=2^{\aleph _{0}}$ , CH can be restated as follows:

Continuum Hypothesis— $\nexists S\colon \aleph _{0}<|S|<2^{\aleph _{0}}$ .

Assuming theaxiom of choice, there is a unique smallest cardinal number $\aleph _{1}$ greater than $\aleph _{0}$ , and the continuum hypothesis is in turn equivalent to the equality $2^{\aleph _{0}}=\aleph _{1}$ .^[6]^[7]

Independence from ZFC

[edit]

The independence of the continuum hypothesis (CH) fromZermelo–Fraenkel set theory (ZF) follows from combined work ofKurt Gödel andPaul Cohen.

Gödel^[8]^[2]showed that CH cannot be disproved from ZF, even if theaxiom of choice (AC) is adopted, i.e. from ZFC. Gödel's proof shows that both CH and AC hold in theconstructible universe $L {\displaystyle L}$ , aninner model of ZF set theory, assuming only the axioms of ZF. The existence of an inner model of ZF in which additional axioms hold shows that the additional axioms are (relatively)consistent with ZF, provided ZF itself is consistent. The latter condition cannot be proved in ZF itself, due toGödel's incompleteness theorems, but is widely believed to be true and can be proved in stronger set theories.

Cohen^[4]^[9] showed that CH cannot be proven from the ZFC axioms, completing the overall independence proof. To prove his result, Cohen developed the method offorcing, which has become a standard tool in set theory. Essentially, this method begins with a model of ZF in which CH holds and constructs another model which contains more sets than the original in a way that CH does not hold in the new model. Cohen was awarded theFields Medal in 1966 for his proof.

Cohen's independence proof shows that CH is independent of ZFC. Further research has shown that CH is independent of all knownlarge cardinal axioms in the context of ZFC.^[10] Moreover, it has been shown that thecardinality of the continuum ${\mathfrak {c}}=2^{\aleph _{0}}$ can be any cardinal consistent withKőnig's theorem. A result of Solovay, proved shortly after Cohen's result on the independence of the continuum hypothesis, shows that in any model of ZFC, if $\kappa$ is a cardinal of uncountablecofinality, then there is a forcing extension in which $2^{\aleph _{0}}=\kappa$ . However, per Kőnig's theorem, it is not consistent to assume $2^{\aleph _{0}}$ is $\aleph _{\omega }$ or $\aleph _{\omega _{1}+\omega }$ or any cardinal with cofinality $\omega$ .

The continuum hypothesis is closely related to many statements inanalysis, point settopology andmeasure theory. As a result of its independence, many substantialconjectures in those fields have subsequently been shown to be independent as well.

The independence from ZFC means that proving or disproving the CH within ZFC is impossible. However, Gödel and Cohen's negative results are not universally accepted as disposing of all interest in the continuum hypothesis. The continuum hypothesis remains an active topic of research: seeWoodin^[11]^[12] andKoellner^[13] for an overview of the current research status.

The continuum hypothesis and theaxiom of choice were among the first genuinely mathematical statements shown to be independent of ZF set theory. Although the existence of some statements independent of ZFC had already been known more than two decades prior: for example, assuminggood soundness properties and the consistency of ZFC,Gödel's incompleteness theorems published in 1931 establish that there is a formal statement Con(ZFC) (one for each appropriateGödel numbering scheme) expressing the consistency of ZFC, that is also independent of it. The latter independence result indeed holds for many theories.

Arguments for and against the continuum hypothesis

[edit]

Gödel believed that CH is false, and that his proof that CH is consistent with ZFC only shows that theZermelo–Fraenkel axioms do not adequately characterize the universe of sets. Gödel was aPlatonist and therefore had no problems with asserting the truth and falsehood of statements independent of their provability. Cohen, though aformalist,^[14] also tended towards rejecting CH.

Historically, mathematicians who favored a "rich" and "large"universe of sets were against CH, while those favoring a "neat" and "controllable" universe favored CH. Parallel arguments were made for and against theaxiom of constructibility, which implies CH. More recently,Matthew Foreman has pointed out thatontological maximalism can actually be used to argue in favor of CH, because among models that have the same reals, models with "more" sets of reals have a better chance of satisfying CH.^[15]

Another viewpoint is that the conception of set is not specific enough to determine whether CH is true or false. This viewpoint was advanced as early as 1923 bySkolem, even before Gödel's first incompleteness theorem. Skolem argued on the basis of what is now known asSkolem's paradox, and it was later supported by the independence of CH from the axioms of ZFC since these axioms are enough to establish the elementary properties of sets and cardinalities. In order to argue against this viewpoint, it would be sufficient to demonstrate new axioms that are supported by intuition and resolve CH in one direction or another. Although theaxiom of constructibility does resolve CH, it is not generally considered to be intuitively true any more than CH is generally considered to be false.^[16]

At least two other axioms have been proposed that have implications for the continuum hypothesis, although these axioms have not currently found wide acceptance in the mathematical community. In 1986, Chris Freiling^[17] presented an argument against CH by showing that the negation of CH is equivalent toFreiling's axiom of symmetry, a statement derived by arguing from particular intuitions aboutprobabilities. Freiling believes this axiom is "intuitively clear"^[17] but others have disagreed.^[18]^[19]

A difficult argument against CH developed byW. Hugh Woodin has attracted considerable attention since the year 2000.^[11]^[12]Foreman does not reject Woodin's argument outright but urges caution.^[20] Woodin proposed a new hypothesis that he labeled the"(*)-axiom", or "Star axiom". The Star axiom would imply that $2^{\aleph _{0}}$ is $\aleph _{2}$ , thus falsifying CH. The Star axiom was bolstered by an independent May 2021 proof showing the Star axiom can be derived from a variation ofMartin's maximum. However, Woodin stated in the 2010s that he now instead believes CH to be true, based on his belief in his new "ultimate L" conjecture.^[21]^[22]

Solomon Feferman argued that CH is not a definite mathematical problem.^[23] He proposed a theory of "definiteness" using a semi-intuitionistic subsystem of ZF that acceptsclassical logic for bounded quantifiers but usesintuitionistic logic for unbounded ones, and suggested that a proposition $\phi$ is mathematically "definite" if the semi-intuitionistic theory can prove $(\phi \lor \neg \phi )$ . He conjectured that CH is not definite according to this notion, and proposed that CH should, therefore, be considered not to have a truth value.Peter Koellner wrote a critical commentary on Feferman's article.^[24]

Joel David Hamkins proposes amultiverse approach to set theory and argues that "the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and, as a result, it can no longer be settled in the manner formerly hoped for".^[25] In a related vein,Saharon Shelah wrote that he does "not agree with the pure Platonic view that the interesting problems in set theory can be decided, that we just have to discover the additional axiom. My mental picture is that we have many possible set theories, all conforming to ZFC".^[26]

Generalized continuum hypothesis

[edit]

Thegeneralized continuum hypothesis (GCH) states that if an infinite set's cardinality lies between that of an infinite setS and that of thepower set ${\mathcal {P}}(S)$ ofS, then it has the same cardinality as eitherS or ${\mathcal {P}}(S)$ . That is, for anyinfinite cardinal $\lambda$ there is no cardinal $\kappa$ such that $\lambda <\kappa <2^{\lambda }$ . GCH is equivalent to:

\aleph _{\alpha +1}=2^{\aleph _{\alpha }}

for everyordinal

\alpha

^[6]

(occasionally calledCantor's aleph hypothesis).

Thebeth numbers provide an alternative notation for this condition: $\aleph _{\alpha }=\beth _{\alpha }$ for every ordinal $\alpha$ . The continuum hypothesis is the special case for the ordinal $\alpha =1$ . GCH was first suggested byPhilip Jourdain.^[27] For the early history of GCH, see Moore.^[28]

Like CH, GCH is also independent of ZFC, butSierpiński proved that ZF + GCH implies theaxiom of choice (AC) (and therefore the negation of theaxiom of determinacy, AD), so choice and GCH are not independent in ZF; there are no models of ZF in which GCH holds and AC fails. To prove this, Sierpiński showed GCH implies that every cardinality n is smaller than somealeph number, and thus can be ordered. This is done by showing that n is smaller than $2^{\aleph _{0}+n}$ which is smaller than its ownHartogs number—this uses the equality $2^{\aleph _{0}+n}\,=\,2\cdot \,2^{\aleph _{0}+n}$ ; for the full proof, see Gillman.^[29]

Kurt Gödel showed that GCH is a consequence of ZF +V=L (the axiom that every set is constructible relative to the ordinals), and is therefore consistent with ZFC. As GCH implies CH, Cohen's model in which CH fails is a model in which GCH fails, and thus GCH is not provable from ZFC. W. B. Easton used the method of forcing developed by Cohen to proveEaston's theorem, which shows it is consistent with ZFC for arbitrarily large cardinals $\aleph _{\alpha }$ to fail to satisfy $2^{\aleph _{\alpha }}=\aleph _{\alpha +1}$ . Much later,Foreman andWoodin proved that (assuming the consistency of very large cardinals) it is consistent that $2^{\kappa }>\kappa ^{+}$ holds for every infinite cardinal $\kappa$ . Later Woodin extended this by showing the consistency of $2^{\kappa }=\kappa ^{++}$ for every $\kappa$ . Carmi Merimovich^[30] showed that, for eachn ≥ 1, it is consistent with ZFC that for each infinite cardinalκ,2^κ is thenth successor ofκ (assuming the consistency of some large cardinal axioms). On the other hand, László Patai^[31] proved that ifγ is an ordinal and for each infinite cardinalκ,2^κ is theγth successor ofκ, thenγ is finite.

For any infinite setsA andB, if there is an injection fromA toB then there is an injection from subsets ofA to subsets ofB. Thus for any infinite cardinalsA andB, $A<B\to 2^{A}\leq 2^{B}$ . IfA andB are finite, the stronger inequality $A<B\to 2^{A}<2^{B}$ holds. GCH implies that this strict, stronger inequality holds for infinite cardinals as well as finite cardinals.

Implications of GCH for cardinal exponentiation

[edit]

Although the generalized continuum hypothesis refers directly only to cardinal exponentiation with 2 as the base, one can deduce from it the values of cardinal exponentiation $\aleph _{\alpha }^{\aleph _{\beta }}$ in all cases. GCH implies that for ordinalsα andβ:^[32]

$\aleph _{\alpha }^{\aleph _{\beta }}=\aleph _{\beta +1}$ whenα ≤β+1;
$\aleph _{\alpha }^{\aleph _{\beta }}=\aleph _{\alpha }$ whenβ+1 <α and $\aleph _{\beta }<\operatorname {cf} (\aleph _{\alpha })$ , wherecf is thecofinality operation; and
$\aleph _{\alpha }^{\aleph _{\beta }}=\aleph _{\alpha +1}$ whenβ+1 <α and $\aleph _{\beta }\geq \operatorname {cf} (\aleph _{\alpha })$ .

The first equality (whenα ≤β+1) follows from:

$\aleph _{\alpha }^{\aleph _{\beta }}\leq \aleph _{\beta +1}^{\aleph _{\beta }}=(2^{\aleph _{\beta }})^{\aleph _{\beta }}=2^{\aleph _{\beta }\cdot \aleph _{\beta }}=2^{\aleph _{\beta }}=\aleph _{\beta +1}$ while:

$\aleph _{\beta +1}=2^{\aleph _{\beta }}\leq \aleph _{\alpha }^{\aleph _{\beta }}.$

The third equality (whenβ+1 <α and $\aleph _{\beta }\geq \operatorname {cf} (\aleph _{\alpha })$ ) follows from:

$\aleph _{\alpha }^{\aleph _{\beta }}\geq \aleph _{\alpha }^{\operatorname {cf} (\aleph _{\alpha })}>\aleph _{\alpha }$

byKőnig's theorem, while:

$\aleph _{\alpha }^{\aleph _{\beta }}\leq \aleph _{\alpha }^{\aleph _{\alpha }}\leq (2^{\aleph _{\alpha }})^{\aleph _{\alpha }}=2^{\aleph _{\alpha }\cdot \aleph _{\alpha }}=2^{\aleph _{\alpha }}=\aleph _{\alpha +1}$

References

[edit]

^Cantor, Georg (1878)."Ein Beitrag zur Mannigfaltigkeitslehre".Journal für die Reine und Angewandte Mathematik.1878 (84):242–258.doi:10.1515/crll.1878.84.242 (inactive 17 December 2025).{{cite journal}}: CS1 maint: DOI inactive as of December 2025 (link)
^^a ^b ^cGödel, Kurt (1940).The Consistency of the Continuum-Hypothesis. Princeton University Press.
^Dauben, Joseph Warren (1990).Georg Cantor: His mathematics and philosophy of the infinite. Princeton University Press. pp. 134–137.ISBN 9780691024479.
^^a ^bCohen, Paul J. (15 December 1963)."The independence of the Continuum Hypothesis, [part I]".Proceedings of the National Academy of Sciences of the United States of America.50 (6):1143–1148.Bibcode:1963PNAS...50.1143C.doi:10.1073/pnas.50.6.1143.JSTOR 71858.PMC 221287.PMID 16578557.
^For a proof sketch, seeCountability of the Rationals
^^a ^bGoldrei, Derek (1996).Classic Set Theory.Chapman & Hall.
^Asaf Karagila (https://math.stackexchange.com/users/622/asaf-karagila), How to formulate continuum hypothesis without the axiom of choice?, URL (version: 2017-04-13):https://math.stackexchange.com/q/404813
^Gödel, Kurt (1938)."The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis".Proceedings of the National Academy of Sciences.24 (12):556–557.Bibcode:1938PNAS...24..556G.doi:10.1073/pnas.24.12.556.PMC 1077160.PMID 16577857.
^Cohen, Paul J. (15 January 1964)."The independence of the Continuum Hypothesis, [part] II".Proceedings of the National Academy of Sciences of the United States of America.51 (1):105–110.Bibcode:1964PNAS...51..105C.doi:10.1073/pnas.51.1.105.JSTOR 72252.PMC 300611.PMID 16591132.
^Feferman, Solomon (February 1999). "Does mathematics need new axioms?".American Mathematical Monthly.106 (2):99–111.CiteSeerX 10.1.1.37.295.doi:10.2307/2589047.JSTOR 2589047.
^^a ^bWoodin, W. Hugh (2001)."The Continuum Hypothesis, Part I"(PDF).Notices of the AMS.48 (6):567–576.Archived(PDF) from the original on 2022-10-10.
^^a ^bWoodin, W. Hugh (2001)."The Continuum Hypothesis, Part II"(PDF).Notices of the AMS.48 (7):681–690.Archived(PDF) from the original on 2022-10-10.
^Koellner, Peter (2011)."The Continuum Hypothesis"(PDF).Exploring the Frontiers of Independence. Harvard lecture series.Archived(PDF) from the original on 2012-01-24.
^Goodman, Nicolas D. (1979). "Mathematics as an objective science".The American Mathematical Monthly.86 (7):540–551.doi:10.2307/2320581.JSTOR 2320581.MR 0542765.This view is often calledformalism. Positions more or less like this may be found in Haskell Curry [5], Abraham Robinson [17], and Paul Cohen [4].
^Maddy 1988, p. 500.
^Kunen, Kenneth (1980).Set Theory: An Introduction to Independence Proofs. Amsterdam, NL: North-Holland. p. 171.ISBN 978-0-444-85401-8.
^^a ^bFreiling, Chris (1986). "Axioms of Symmetry: Throwing darts at the real number line".Journal of Symbolic Logic.51 (1). Association for Symbolic Logic:190–200.doi:10.2307/2273955.JSTOR 2273955.S2CID 38174418.
^Bagemihl, F. (1989–1990)."Throwing a dart at Freiling's argument against the continuum hypothesis".Real Analysis Exchange.15 (1):342–345.doi:10.2307/44152014.JSTOR 44152014.MR 1042552.
^Hamkins, Joel David (January 2015). "Is the Dream Solution of the Continuum Hypothesis Attainable?".Notre Dame Journal of Formal Logic.56 (1).arXiv:1203.4026.doi:10.1215/00294527-2835047.
^Foreman, Matt (2003)."Has the Continuum Hypothesis been settled?"(PDF).Archived(PDF) from the original on 2022-10-10. Retrieved25 February 2006.
^Wolchover, Natalie (15 July 2021)."How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer".Quanta Magazine. Retrieved30 December 2021.
^Rittberg, Colin J. (March 2015). "How Woodin changed his mind: new thoughts on the Continuum Hypothesis".Archive for History of Exact Sciences.69 (2):125–151.doi:10.1007/s00407-014-0142-8.S2CID 122205863.
^Feferman, Solomon (2011)."Is the Continuum Hypothesis a definite mathematical problem?"(PDF).Exploring the Frontiers of Independence. Harvard lecture series.Archived(PDF) from the original on 2022-10-10.
^Koellner, Peter (2011)."Feferman on the indefiniteness of CH"(PDF).Archived(PDF) from the original on 2012-03-19.
^Hamkins, Joel David (2012). "The set-theoretic multiverse".The Review of Symbolic Logic.5 (3):416–449.arXiv:1108.4223.doi:10.1017/S1755020311000359.S2CID 33807508.
^Shelah, Saharon (2003). "Logical dreams".Bulletin of the American Mathematical Society. New Series.40 (2):203–228.arXiv:math/0211398.doi:10.1090/s0273-0979-03-00981-9.S2CID 1510438.
^Jourdain, Philip E.B. (1905)."On transfinite cardinal numbers of the exponential form".Philosophical Magazine. Series 6.9 (49):42–56.doi:10.1080/14786440509463254.
^Moore, Gregory H. (2011). "Early history of the generalized continuum hypothesis: 1878–1938".Bulletin of Symbolic Logic.17 (4):489–532.doi:10.2178/bsl/1318855631.MR 2896574.
^Gillman, Leonard (2002)."Two classical surprises concerning the Axiom of Choice and the Continuum Hypothesis"(PDF).American Mathematical Monthly.109 (6):544–553.doi:10.2307/2695444.JSTOR 2695444.Archived(PDF) from the original on 2022-10-10.
^Merimovich, Carmi (2007). "A power function with a fixed finite gap everywhere".Journal of Symbolic Logic.72 (2):361–417.arXiv:math/0005179.doi:10.2178/jsl/1185803615.MR 2320282.S2CID 15577499.
^Patai, L. (1930). "Untersuchungen über die א-reihe".Mathematische und naturwissenschaftliche Berichte aus Ungarn (in German).37:127–142.
^Hayden, Seymour; Kennison, John F. (1968).Zermelo-Fraenkel Set Theory. Columbus, Ohio: Charles E. Merrill. p. 147, exercise 76.

Maddy, Penelope (June 1988). "Believing the axioms, [part I]".Journal of Symbolic Logic.53 (2). Association for Symbolic Logic:481–511.doi:10.2307/2274520.JSTOR 2274520.

Sources

[edit]

This article incorporates material from Generalized continuum hypothesis onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.Archived 2017-02-08 at theWayback Machine

External links

[edit]

Quotations related toContinuum hypothesis at Wikiquote

Szudzik, Matthew &Weisstein, Eric W."Continuum Hypothesis".MathWorld.

v t e Set theory
Overview	Set (mathematics)
Axioms	Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity Union Martin's axiom Axiom schema replacement specification
Operations	Cartesian product Complement (i.e. set difference) De Morgan's laws Disjoint union Identities Intersection Power set Symmetric difference Union
Concepts Methods	Almost Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram
Set types	Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset · Superset Transitive Uncountable Universal
Theories	Alternative Axiomatic Naive Cantor's theorem Zermelo General Principia Mathematica New Foundations Zermelo–Fraenkel von Neumann–Bernays–Gödel Morse–Kelley Kripke–Platek Tarski–Grothendieck
Paradoxes Problems	Russell's paradox Suslin's problem Burali-Forti paradox
Set theorists	Paul Bernays Georg Cantor Paul Cohen Richard Dedekind Abraham Fraenkel Kurt Gödel Thomas Jech John von Neumann Willard Quine Bertrand Russell Thoralf Skolem Ernst Zermelo

Mathematical logic

General

Theorems
(list),
paradoxes

Logics

Traditional	Classical logic Logical truth Tautology Proposition Inference Logical equivalence Consistency Equiconsistency Argument Soundness Validity Syllogism Square of opposition Venn diagram
Propositional	Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued logic 3 finite ∞
Predicate	First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic predicate calculus

Set theory

Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian product power set identities
Types ofsets	Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann
Maps, cardinality	Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering Enumeration Large cardinal inaccessible Aleph number Operation binary
Theories	Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck Von Neumann–Bernays–Gödel Ackermann Constructive