Movatterモバイル変換

[0]ホーム
URL:

Jump to content
WikipediaThe Free Encyclopedia
Search

Zermelo–Fraenkel set theory

From Wikipedia, the free encyclopedia
Standard system of axiomatic set theory
"ZFC" redirects here. For other uses, seeZFC (disambiguation).
This article includes a list ofgeneral references, butit lacks sufficient correspondinginline citations. Please help toimprove this article byintroducing more precise citations.(February 2024) (Learn how and when to remove this message)

Inset theory,Zermelo–Fraenkel set theory, named after mathematiciansErnst Zermelo andAbraham Fraenkel, is anaxiomatic system that was proposed in the early twentieth century in order to formulate atheory of sets free of paradoxes such asRussell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversialaxiom of choice (AC) included, is the standard form ofaxiomatic set theory and as such is the most commonfoundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviatedZFC, where C stands for "choice",[1] andZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.

Informally,[2] Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of ahereditarywell-foundedset, so that allentities in theuniverse of discourse are such sets. Thus theaxioms of Zermelo–Fraenkel set theory refer only topure sets and prevent itsmodels from containingurelements (elements that are not themselves sets). Furthermore,proper classes (collections ofmathematical objects defined by a property shared by their members where the collections are too big to be sets) can only be treated indirectly. Specifically, Zermelo–Fraenkel set theory does not allow for the existence of auniversal set (a set containing all sets) nor forunrestricted comprehension, thereby avoiding Russell's paradox.Von Neumann–Bernays–Gödel set theory (NBG) is a commonly usedconservative extension of Zermelo–Fraenkel set theory that does allow explicit treatment of proper classes.

There are many equivalent formulations of the axioms of Zermelo–Fraenkel set theory. Most of the axioms state the existence of particular sets defined from other sets. For example, theaxiom of pairing implies that given any two setsa{\displaystyle a} andb{\displaystyle b} there is a new set{a,b}{\displaystyle \{a,b\}} containing exactlya{\displaystyle a} andb{\displaystyle b}.[a] Other axioms describe properties of set membership. A goal of the axioms is that each axiom should be true if interpreted as a statement about the collection of all sets in thevon Neumann universe (also known as the cumulative hierarchy).

Themetamathematics of Zermelo–Fraenkel set theory has been extensively studied. Landmark results in this area established thelogical independence of the axiom of choice from the remaining Zermelo-Fraenkel axioms and of thecontinuum hypothesis from ZFC. Theconsistency of a theory such as ZFC cannot be proved within the theory itself, as shown byGödel's second incompleteness theorem.

History

[edit]
Main article:History of set theory

The modern study ofset theory was initiated byGeorg Cantor andRichard Dedekind in the 1870s. However, the discovery ofparadoxes innaive set theory, such asRussell's paradox, led to the desire for a more rigorous form of set theory that was free of these paradoxes.

In 1908,Ernst Zermelo proposed the firstaxiomatic set theory,Zermelo set theory. However, as first pointed out byAbraham Fraenkel in a 1921 letter to Zermelo, this theory was incapable of proving the existence of certain sets andcardinal numbers whose existence was taken for granted by most set theorists of the time, notably the cardinal numberaleph-omega (ℵω{\displaystyle \aleph _{\omega }}) and the set{Z0,P(Z0),P(P(Z0)),P(P(P(Z0))),...},{\displaystyle \{Z_{0},{\mathcal {P}}(Z_{0}),{\mathcal {P}}({\mathcal {P}}(Z_{0})),{\mathcal {P}}({\mathcal {P}}({\mathcal {P}}(Z_{0}))),...\},} whereZ0{\displaystyle Z_{0}} is any infinite set andP{\displaystyle {\mathcal {P}}} is thepower set operation.[3] Moreover, one of Zermelo's axioms invoked a concept, that of a "definite" property, whose operational meaning was not clear. In 1922, Fraenkel andThoralf Skolem independently proposed operationalizing a "definite" property as one that could be formulated as a well-formed formula in afirst-order logic whoseatomic formulas were limited to set membership and identity. They also independently proposed replacing theaxiom schema of specification with theaxiom schema of replacement. Appending this schema, as well as theaxiom of regularity (first proposed byJohn von Neumann),[4] to Zermelo set theory yields the theory ZFC.

Formal language

[edit]
See also:Formal language

Formally, ZFC is aone-sorted theory infirst-order logic. The equality symbol can be treated as either a primitive logical symbol or a high-level abbreviation for having exactly the same elements. The former approach is the most common. Thesignature has a single predicate symbol, usually denoted∈{\displaystyle \in }, which is a predicate symbol of arity 2 (a binary relation symbol). This symbol symbolizes aset membership relation. For example, theformulaa∈b{\displaystyle a\in b} means thata{\displaystyle a} is an element of the setb{\displaystyle b} (also read asa{\displaystyle a} is a member ofb{\displaystyle b}).

There are different ways to formulate the formal language. Some authors may choose a different set of connectives or quantifiers. For example, the logical connective NAND alone can encode the other connectives, a property known asfunctional completeness. This section attempts to strike a balance between simplicity and intuitiveness.

The language's alphabet consists of:

With this alphabet, the recursive rules for formingwell-formed formulae (wff) are as follows:

x=y{\displaystyle x=y}
x∈y{\displaystyle x\in y}
¬ϕ{\displaystyle \lnot \phi }
(ϕ∧ψ){\displaystyle (\phi \land \psi )}
(ϕ∨ψ){\displaystyle (\phi \lor \psi )}
∀xϕ{\displaystyle \forall x\phi }
∃xϕ{\displaystyle \exists x\phi }

A well-formed formula can be thought as a syntax tree. The leaf nodes are always atomic formulae. Nodes∧{\displaystyle \land } and∨{\displaystyle \lor } have exactly two child nodes, while nodes¬{\displaystyle \lnot },∀x{\displaystyle \forall x} and∃x{\displaystyle \exists x} have exactly one. There are countably infinitely many wffs, however, each wff has a finite number of nodes.

Axioms

[edit]
See also:Axiom

There are many equivalent formulations of the ZFC axioms.[5] The following particular axiom set is fromKunen (1980). The axioms in order below are expressed in a mixture offirst-order logic and high-level abbreviations.

Axioms 1–8 form ZF, while the axiom 9 turns ZF into ZFC. FollowingKunen (1980), we use the equivalentwell-ordering theorem in place of theaxiom of choice for axiom 9.

All formulations of ZFC imply that at least one set exists. Kunen includes an axiom that directly asserts the existence of a set, although he notes that he does so only "for emphasis".[6] Its omission here can be justified in two ways. First, in the standard semantics of first-order logic in which ZFC is typically formalized, thedomain of discourse must be nonempty. Hence, it is a logical theorem of first-order logic that something exists – usually expressed as the assertion that something is identical to itself,∃x(x=x){\displaystyle \exists x(x=x)}. Consequently, it is a theorem of every first-order theory that something exists. However, as noted above, because in the intended semantics of ZFC, there are only sets, the interpretation of this logical theorem in the context of ZFC is that someset exists. Hence, there is no need for a separate axiom asserting that a set exists. Second, however, even if ZFC is formulated in so-calledfree logic, in which it is not provable from logic alone that something exists, the axiom of infinity asserts that aninfinite set exists. This implies thata set exists, and so, once again, it is superfluous to include an axiom asserting as much.

Axiom of extensionality

[edit]
Main article:Axiom of extensionality

Two sets are equal (are the same set) if they have the same elements.

∀x∀y[∀z(z∈x⇔z∈y)⇒x=y].{\displaystyle \forall x\forall y[\forall z(z\in x\Leftrightarrow z\in y)\Rightarrow x=y].}

The converse of this axiom follows from the substitution property ofequality. ZFC is constructed in first-order logic. Some formulations of first-order logic include identity; others do not. If the variety of first-order logic in which one is constructing set theory does not include equality "={\displaystyle =}",x=y{\displaystyle x=y} may be defined as an abbreviation for the following formula:[7]∀z[z∈x⇔z∈y]∧∀w[x∈w⇔y∈w].{\displaystyle \forall z[z\in x\Leftrightarrow z\in y]\land \forall w[x\in w\Leftrightarrow y\in w].}

In this case, the axiom of extensionality can be reformulated as

∀x∀y[∀z(z∈x⇔z∈y)⇒∀w(x∈w⇔y∈w)],{\displaystyle \forall x\forall y[\forall z(z\in x\Leftrightarrow z\in y)\Rightarrow \forall w(x\in w\Leftrightarrow y\in w)],}

which says that ifx{\displaystyle x} andy{\displaystyle y} have the same elements, then they belong to the same sets.[8]

Axiom of regularity (also called the axiom of foundation)

[edit]
Main article:Axiom of regularity

Every non-empty setx{\displaystyle x} contains a membery{\displaystyle y} such thatx{\displaystyle x} andy{\displaystyle y} aredisjoint sets.

∀x[(∃a(a∈x))⇒∃y(y∈x∧¬∃z(z∈y∧z∈x))].{\displaystyle \forall x[(\exists a(a\in x))\Rightarrow \exists y(y\in x\land \lnot \exists z(z\in y\land z\in x))].}[9]

or in modern notation:∀x(x≠∅⇒∃y(y∈x∧y∩x=∅)).{\displaystyle \forall x\,(x\neq \varnothing \Rightarrow \exists y(y\in x\land y\cap x=\varnothing )).}

This (along with the axioms of pairing and union) implies, for example, that no set is an element of itself and that every set has anordinalrank.

Axiom schema of specification (or of separation, or of restricted comprehension)

[edit]
Main article:Axiom schema of specification

Subsets are commonly constructed usingset builder notation. For example, the even integers can be constructed as the subset of the integersZ{\displaystyle \mathbb {Z} } satisfying thecongruence modulo predicatex≡0(mod2){\displaystyle x\equiv 0{\pmod {2}}}:

{x∈Z:x≡0(mod2)}.{\displaystyle \{x\in \mathbb {Z} :x\equiv 0{\pmod {2}}\}.}

In general, the subset of a setz{\displaystyle z} obeying a formulaφ(x){\displaystyle \varphi (x)} with one free variablex{\displaystyle x} may be written as:

{x∈z:φ(x)}.{\displaystyle \{x\in z:\varphi (x)\}.}

The axiom schema of specification states that this subset always exists (it is anaxiomschema because there is one axiom for eachφ{\displaystyle \varphi }). Formally, letφ{\displaystyle \varphi } be any formula in the language of ZFC with all free variables amongx,z,w1,…,wn{\displaystyle x,z,w_{1},\ldots ,w_{n}} (y{\displaystyle y} is not free inφ{\displaystyle \varphi }). Then:

∀z∀w1∀w2…∀wn∃y∀x[x∈y⇔((x∈z)∧φ(x,w1,w2,...,wn,z))].{\displaystyle \forall z\forall w_{1}\forall w_{2}\ldots \forall w_{n}\exists y\forall x[x\in y\Leftrightarrow ((x\in z)\land \varphi (x,w_{1},w_{2},...,w_{n},z))].}

Note that the axiom schema of specification can only construct subsets and does not allow the construction of entities of the more general form:

{x:φ(x)}.{\displaystyle \{x:\varphi (x)\}.}

This restriction is necessary to avoidRussell's paradox (lety={x:x∉x}{\displaystyle y=\{x:x\notin x\}} theny∈y⇔y∉y{\displaystyle y\in y\Leftrightarrow y\notin y}) and its variants that accompany naive set theory withunrestricted comprehension (since under this restrictiony{\displaystyle y} only refers to setswithinz{\displaystyle z} that don't belong to themselves, andy∈z{\displaystyle y\in z} hasnot been established, even thoughy⊆z{\displaystyle y\subseteq z} is the case, soy{\displaystyle y} stands in a separate position from which it can't refer to or comprehend itself; therefore, in a certain sense, this axiom schema is saying that in order to build ay{\displaystyle y} on the basis of a formulaφ(x){\displaystyle \varphi (x)}, we need to previously restrict the setsy{\displaystyle y} will regard within a setz{\displaystyle z} that leavesy{\displaystyle y} outside soy{\displaystyle y} can't refer to itself; or, in other words, sets shouldn't refer to themselves).

In some other axiomatizations of ZF, this axiom is redundant in that it follows from theaxiom schema of replacement and theaxiom of the empty set.

On the other hand, the axiom schema of specification can be used to prove the existence of theempty set, denoted∅{\displaystyle \varnothing }, once at least one set is known to exist. One way to do this is to use a propertyφ{\displaystyle \varphi } which no set has. For example, ifw{\displaystyle w} is any existing set, the empty set can be constructed as

∅={u∈w∣(u∈u)∧¬(u∈u)}.{\displaystyle \varnothing =\{u\in w\mid (u\in u)\land \lnot (u\in u)\}.}

Thus, theaxiom of the empty set is implied by the nine axioms presented here. The axiom of extensionality implies the empty set is unique (does not depend onw{\displaystyle w}). It is common to make adefinitional extension that adds the symbol "∅{\displaystyle \varnothing }" to the language of ZFC.

Axiom of pairing

[edit]
Main article:Axiom of pairing

Ifx{\displaystyle x} andy{\displaystyle y} are sets, then there exists a set which containsx{\displaystyle x} andy{\displaystyle y} as elements; for example, ifx={1,2}{\displaystyle x=\{1,2\}} andy={2,3}{\displaystyle y=\{2,3\}}, thenz{\displaystyle z} might be{{1,2},{2,3}}{\displaystyle \{\{1,2\},\{2,3\}\}}.

∀x∀y∃z((x∈z)∧(y∈z)).{\displaystyle \forall x\forall y\exists z((x\in z)\land (y\in z)).}

The axiom schema of specification must be used to reduce this to a set with exactly these two elements.

Axiom of union

[edit]
Main article:Axiom of union

Theunion over the elements of a set exists. For example, the union over the elements of the set{{1,2},{2,3}}{\displaystyle \{\{1,2\},\{2,3\}\}} is{1,2,3}.{\displaystyle \{1,2,3\}.}

The axiom of union states that for any set of setsF{\displaystyle {\mathcal {F}}}, there is a setA{\displaystyle A} containing every element that is a member of some member ofF{\displaystyle {\mathcal {F}}}:

∀F∃A∀Y∀x[(x∈Y∧Y∈F)⇒x∈A].{\displaystyle \forall {\mathcal {F}}\,\exists A\,\forall Y\,\forall x[(x\in Y\land Y\in {\mathcal {F}})\Rightarrow x\in A].}

Although this formula doesn't directly assert the existence of∪F{\displaystyle \cup {\mathcal {F}}}, the set∪F{\displaystyle \cup {\mathcal {F}}} can be constructed fromA{\displaystyle A} in the above using the axiom schema of specification:

∪F={x∈A:∃Y(x∈Y∧Y∈F)}.{\displaystyle \cup {\mathcal {F}}=\{x\in A:\exists Y(x\in Y\land Y\in {\mathcal {F}})\}.}

Axiom schema of replacement

[edit]
Main article:Axiom schema of replacement

The axiom schema of replacement asserts that theimage of a set under any definablefunction will also fall inside a set.

Formally, letφ{\displaystyle \varphi } be anyformula in the language of ZFC whosefree variables are amongx,y,A,w1,…,wn,{\displaystyle x,y,A,w_{1},\dotsc ,w_{n},} so that in particularB{\displaystyle B} is not free inφ{\displaystyle \varphi }. Then:

∀A∀w1∀w2…∀wn[∀x(x∈A⇒∃!yφ)⇒∃B ∀x(x∈A⇒∃y(y∈B∧φ))].{\displaystyle \forall A\forall w_{1}\forall w_{2}\ldots \forall w_{n}{\bigl [}\forall x(x\in A\Rightarrow \exists !y\,\varphi )\Rightarrow \exists B\ \forall x{\bigl (}x\in A\Rightarrow \exists y(y\in B\land \varphi ){\bigr )}{\bigr ]}.}

(Theunique existential quantifier∃!{\displaystyle \exists !} denotes the existence of exactly one element such that it follows a given statement.)

In other words, if the relationφ{\displaystyle \varphi } represents a definable functionf{\displaystyle f},A{\displaystyle A} represents itsdomain, andf(x){\displaystyle f(x)} is a set for everyx∈A,{\displaystyle x\in A,} then therange off{\displaystyle f} is a subset of some setB{\displaystyle B}. The form stated here, in whichB{\displaystyle B} may be larger than strictly necessary, is sometimes called theaxiom schema of collection.

Axiom of infinity

[edit]
Main article:Axiom of infinity
First several von Neumann ordinals
0={}=∅
1={0}={∅}
2={0,1}={∅,{∅}}
3={0,1,2}={∅,{∅},{∅,{∅}}}
4={0,1,2,3}={∅,{∅},{∅,{∅}},{∅,{∅},{∅,{∅}}}}

LetS(w){\displaystyle S(w)} abbreviatew∪{w},{\displaystyle w\cup \{w\},} wherew{\displaystyle w} is some set. (We can see that{w}{\displaystyle \{w\}} is a valid set by applying the axiom of pairing withx=y=w{\displaystyle x=y=w} so that the setz is{w}{\displaystyle \{w\}}). Then there exists a setX such that the empty set∅{\displaystyle \varnothing }, defined axiomatically, is a member ofX and, whenever a sety is a member ofX thenS(y){\displaystyle S(y)} is also a member ofX.

∃X[∃e(∀z¬(z∈e)∧e∈X)∧∀y(y∈X⇒S(y)∈X)].{\displaystyle \exists X\left[\exists e(\forall z\,\neg (z\in e)\land e\in X)\land \forall y(y\in X\Rightarrow S(y)\in X)\right].}

or in modern notation:∃X[∅∈X∧∀y(y∈X⇒S(y)∈X)].{\displaystyle \exists X\left[\varnothing \in X\land \forall y(y\in X\Rightarrow S(y)\in X)\right].}

More colloquially, there exists a setX having infinitely many members. These members are built by repeatedly applying the operationS{\displaystyle S} starting from the empty set. Each result of this construction is distinct from the previous ones, so the process does not loop or repeat. The minimal setX satisfying the axiom of infinity is thevon Neumann ordinalω, which can also be thought of as the set ofnatural numbersN{\displaystyle \mathbb {N} }. (Note that the well-foundedness ofN{\displaystyle \mathbb {N} } does not require the axiom of regularity; it follows naturally from the structure of the construction.)

Axiom of power set

[edit]
Main article:Axiom of power set

By definition, a setz{\displaystyle z} is asubset of a setx{\displaystyle x} if and only if every element ofz{\displaystyle z} is also an element ofx{\displaystyle x}:

(z⊆x)⇔(∀q(q∈z⇒q∈x)).{\displaystyle (z\subseteq x)\Leftrightarrow (\forall q(q\in z\Rightarrow q\in x)).}

The Axiom of power set states that for any setx{\displaystyle x}, there is a sety{\displaystyle y} that contains every subset ofx{\displaystyle x}:

∀x∃y∀z(z⊆x⇒z∈y).{\displaystyle \forall x\exists y\forall z(z\subseteq x\Rightarrow z\in y).}

The axiom schema of specification is then used to define thepower setP(x){\displaystyle {\mathcal {P}}(x)} as the subset of such ay{\displaystyle y} containing the subsets ofx{\displaystyle x} exactly:

P(x)={z∈y:z⊆x}.{\displaystyle {\mathcal {P}}(x)=\{z\in y:z\subseteq x\}.}

Axioms1–8 define ZF. Alternative forms of these axioms are often encountered, some of which are listed inJech (2003). Some ZF axiomatizations include an axiom asserting that theempty set exists. The axioms of pairing, union, replacement, and power set are often stated so that the members of the setx{\displaystyle x} whose existence is being asserted are just those sets which the axiom assertsx{\displaystyle x} must contain.

The following axiom is added to turn ZF into ZFC:

Axiom of well-ordering (choice)

[edit]
Main articles:Axiom of choice,Well-ordering theorem, andZorn's lemma

The last axiom, commonly known as theaxiom of choice, is presented here as a property aboutwell-orders, as inKunen (1980).For any setX{\displaystyle X}, there exists abinary relationR{\displaystyle R} whichwell-ordersX{\displaystyle X}. This meansR{\displaystyle R} is alinear order onX{\displaystyle X} such that every nonemptysubset ofX{\displaystyle X} has aleast element under the orderR{\displaystyle R}.

∀X∃R(Rwell-ordersX).{\displaystyle \forall X\exists R(R\;{\mbox{well-orders}}\;X).}

Given axioms1 – 8, many statements are provably equivalent to axiom9. The most common of these goes as follows. LetX{\displaystyle X} be a set whose members are all nonempty. Then there exists a functionf{\displaystyle f} fromX{\displaystyle X} to the union of the members ofX{\displaystyle X}, called a "choice function", such that for allY∈X{\displaystyle Y\in X} one hasf(Y)∈Y{\displaystyle f(Y)\in Y}. A third version of the axiom, also equivalent, isZorn's lemma.

Since the existence of a choice function whenX{\displaystyle X} is afinite set is easily proved from axioms1–8, AC only matters for certaininfinite sets. AC is characterized asnonconstructive because it asserts the existence of a choice function but says nothing about how this choice function is to be "constructed".

Motivation via the cumulative hierarchy

[edit]
Further information:Von Neumann universe

One motivation for the ZFC axioms isthe cumulative hierarchy of sets introduced byJohn von Neumann.[10] In this viewpoint, the universe of set theory is built up in stages, with one stage for eachordinal number. At stage 0, there are no sets yet. At each following stage, a set is added to the universe if all of its elements have been added at previous stages. Thus the empty set is added at stage 1, and the set containing the empty set is added at stage 2.[11] The collection of all sets that are obtained in this way, over all the stages, is known asV. The sets inV can be arranged into a hierarchy by assigning to each set the first stage at which that set was added toV.

It is provable that a set is inV if and only if the set ispure andwell-founded. AndV satisfies all the axioms of ZFC if the class of ordinals has appropriate reflection properties. For example, suppose that a setx is added at stage α, which means that every element ofx was added at a stage earlier than α. Then, every subset ofx is also added at (or before) stage α, because all elements of any subset ofx were also added before stage α. This means that any subset ofx which the axiom of separation can construct is added at (or before) stage α, and that the powerset ofx will be added at the next stage after α.[12]

The picture of the universe of sets stratified into the cumulative hierarchy is characteristic of ZFC and related axiomatic set theories such asVon Neumann–Bernays–Gödel set theory (often called NBG) andMorse–Kelley set theory. The cumulative hierarchy is not compatible with other set theories such asNew Foundations.

It is possible to change the definition ofV so that at each stage, instead of adding all the subsets of the union of the previous stages, subsets are only added if they are definable in a certain sense. This results in a more "narrow" hierarchy, which gives theconstructible universeL, which also satisfies all the axioms of ZFC, including the axiom of choice. It is independent from the ZFC axioms whetherV = L. Although the structure ofL is more regular and well behaved than that of V, few mathematicians argue that V = L should be added to ZFC as an additional "axiom of constructibility".

Metamathematics

[edit]

Virtual classes

[edit]

Proper classes (collections of mathematical objects defined by a property shared by their members which are too big to be sets) can only be treated indirectly in ZF (and thus ZFC).An alternative to proper classes while staying within ZF and ZFC is thevirtual class notational construct introduced byQuine (1969), where the entire constructy ∈ {x | Fx } is simply defined as Fy.[13] This provides a simple notation for classes that can contain sets but need not themselves be sets, while not committing to the ontology of classes (because the notation can be syntactically converted to one that only uses sets). Quine's approach built on the earlier approach ofBernays & Fraenkel (1958). Virtual classes are also used inLevy (2002),Takeuti & Zaring (1982), and in theMetamath implementation of ZFC.

Finite axiomatization

[edit]
Main article:Von Neumann–Bernays–Gödel set theory

The axiom schemata of replacement and separation each contain infinitely many instances.Montague (1961) included a result first proved in his 1957 Ph.D. thesis: if ZFC is consistent, it is impossible to axiomatize ZFC using only finitely many axioms. On the other hand,von Neumann–Bernays–Gödel set theory (NBG) can be finitely axiomatized. The ontology of NBG includesproper classes as well as sets; a set is any class that can be a member of another class. NBG and ZFC are equivalent set theories in the sense that anytheorem not mentioning classes and provable in one theory can be proved in the other.

Consistency

[edit]

Gödel's second incompleteness theorem says that a recursively axiomatizable system that can interpretRobinson arithmetic can prove its own consistency only if it is inconsistent. Moreover, Robinson arithmetic can be interpreted ingeneral set theory, a small fragment of ZFC. Hence theconsistency of ZFC cannot be proved within ZFC itself (unless it is actually inconsistent). Thus, to the extent that ZFC is identified with ordinary mathematics, the consistency of ZFC cannot be demonstrated in ordinary mathematics. The consistency of ZFC does follow from the existence of a weaklyinaccessible cardinal, which is unprovable in ZFC if ZFC is consistent. Nevertheless, it is deemed unlikely that ZFC harbors an unsuspected contradiction; it is widely believed that if ZFC were inconsistent, that fact would have been uncovered by now. This much is certain – ZFC is immune to the classic paradoxes ofnaive set theory:Russell's paradox, theBurali-Forti paradox, andCantor's paradox.

Abian & LaMacchia (1978) studied asubtheory of ZFC consisting of the axioms of extensionality, union, powerset, replacement, and choice. Usingmodels, they proved this subtheory consistent, and proved that each of the axioms of extensionality, replacement, and power set is independent of the four remaining axioms of this subtheory. If this subtheory is augmented with the axiom of infinity, each of the axioms of union, choice, and infinity is independent of the five remaining axioms. Because there are non-well-founded models that satisfy each axiom of ZFC except the axiom of regularity, that axiom is independent of the other ZFC axioms.

If consistent, ZFC cannot prove the existence of theinaccessible cardinals thatcategory theory requires. Huge sets of this nature are possible if ZF is augmented withTarski's axiom.[14] Assuming that axiom turns the axioms ofinfinity,power set, andchoice (7 – 9 above) into theorems.

Independence

[edit]

Many important statements areindependentof ZFC. The independence is usually proved byforcing, whereby it is shown that every countable transitivemodel of ZFC (sometimes augmented withlarge cardinal axioms) can be expanded to satisfy the statement in question. A different expansion is then shown to satisfy the negation of the statement. An independence proof by forcing automatically proves independence from arithmetical statements, other concrete statements, and large cardinal axioms. Some statements independent of ZFC can be proven to hold in particularinner models, such as in theconstructible universe. However, some statements that are true about constructible sets are not consistent with hypothesized large cardinal axioms.

Forcing proves that the following statements are independent of ZFC:

Remarks:

A variation on the method offorcing can also be used to demonstrate the consistency and unprovability of theaxiom of choice, i.e., that the axiom of choice is independent of ZF. The consistency of choice can be (relatively) easily verified by proving that the inner model L satisfies choice. (Thus every model of ZF contains a submodel of ZFC, so that Con(ZF) implies Con(ZFC).) Since forcing preserves choice, we cannot directly produce a model contradicting choice from a model satisfying choice. However, we can use forcing to create a model which contains a suitable submodel, namely one satisfying ZF but not C.

Another method of proving independence results, one owing nothing to forcing, is based on Gödel's second incompleteness theorem. This approach employs the statement whose independence is being examined, to prove the existence of a set model of ZFC, in which case Con(ZFC) is true. Since ZFC satisfies the conditions of Gödel's second theorem, the consistency of ZFC is unprovable in ZFC (provided that ZFC is, in fact, consistent). Hence no statement allowing such a proof can be proved in ZFC. This method can prove that the existence oflarge cardinals is not provable in ZFC, but cannot prove that assuming such cardinals, given ZFC, is free of contradiction.

Proposed additions

[edit]

The project to unify set theorists behind additional axioms to resolve the continuum hypothesis or other meta-mathematical ambiguities is sometimes known as "Gödel's program".[15] Mathematicians currently debate which axioms are the most plausible or "self-evident", which axioms are the most useful in various domains, and about to what degree usefulness should be traded off with plausibility; some "multiverse" set theorists argue that usefulness should be the sole ultimate criterion in which axioms to customarily adopt. One school of thought leans on expanding the "iterative" concept of a set to produce a set-theoretic universe with an interesting and complex but reasonably tractable structure by adopting forcing axioms; another school advocates for a tidier, less cluttered universe, perhaps focused on a "core" inner model.[16]

Criticisms

[edit]
See also:projective determinacy

ZFC has been criticized both for being excessively strong and for being excessively weak, as well as for its failure to capture objects such as proper classes and theuniversal set.

Many mathematical theorems can be proven in much weaker systems than ZFC, such asPeano arithmetic andsecond-order arithmetic (as explored by the program ofreverse mathematics).Saunders Mac Lane andSolomon Feferman have both made this point. Some of "mainstream mathematics" (mathematics not directly connected with axiomatic set theory) is beyond Peano arithmetic and second-order arithmetic, but still, all such mathematics can be carried out in ZC (Zermelo set theory with choice), another theory weaker than ZFC. Much of the power of ZFC, including the axiom of regularity and the axiom schema of replacement, is included primarily to facilitate the study of the set theory itself.

On the other hand, amongaxiomatic set theories, ZFC is comparatively weak. UnlikeNew Foundations, ZFC does not admit the existence of a universal set. Hence theuniverse of sets under ZFC is not closed under the elementary operations of thealgebra of sets. Unlikevon Neumann–Bernays–Gödel set theory (NBG) andMorse–Kelley set theory (MK), ZFC does not admit the existence ofproper classes. A further comparative weakness of ZFC is that theaxiom of choice included in ZFC is weaker than theaxiom of global choice included in NBG and MK.

There are numerousmathematical statements independent of ZFC. These include thecontinuum hypothesis, theWhitehead problem, and thenormal Moore space conjecture. Some of these conjectures are provable with the addition of axioms such asMartin's axiom orlarge cardinal axioms to ZFC. Some others are decided in ZF+AD where AD is theaxiom of determinacy, a strong supposition incompatible with choice. One attraction of large cardinal axioms is that they enable many results from ZF+AD to be established in ZFC adjoined by some large cardinal axiom. TheMizar system andmetamath have adoptedTarski–Grothendieck set theory, an extension of ZFC, so that proofs involvingGrothendieck universes (encountered in category theory and algebraic geometry) can be formalized.

See also

[edit]

Relatedaxiomatic set theories:

Notes

[edit]
  1. ^At least in Kunen's formulation, the exact statement of the axiom is that there is a set containing botha{\displaystyle a} andb{\displaystyle b}. To conclude that a set containingexactlya{\displaystyle a} andb{\displaystyle b} exists requires an application of theaxiom schema of separation.

References

[edit]
  1. ^Ciesielski 1997, p. 4: "Zermelo-Fraenkel axioms (abbreviated as ZFC where C stands for the axiom of Choice)"
  2. ^Kunen 2007, p. 10
  3. ^Ebbinghaus 2007, p. 136.
  4. ^Halbeisen 2011, pp. 62–63.
  5. ^Fraenkel, Bar-Hillel & Lévy 1973
  6. ^Kunen 1980, p. 10.
  7. ^Hatcher 1982, p. 138, def. 1.
  8. ^Fraenkel, Bar-Hillel & Lévy 1973.
  9. ^Shoenfield 2001, p. 239.
  10. ^Shoenfield 1977, section 2.
  11. ^Hinman 2005, p. 467.
  12. ^For a complete argument thatV satisfies ZFC seeShoenfield (1977).
  13. ^Link 2014
  14. ^Tarski 1939.
  15. ^Feferman 1996.
  16. ^Wolchover 2013.

Bibliography

[edit]

External links

[edit]
Overview
Venn diagram of set intersection
Axioms
Operations
  • Concepts
  • Methods
Set types
Theories
Set theorists
General
Theorems (list)
 and paradoxes
Logics
Traditional
Propositional
Predicate
Set theory
Types ofsets
Maps and cardinality
Set theories
Formal systems (list),
language and syntax
Example axiomatic
systems (list)
Proof theory
Model theory
Computability theory
Related
Retrieved from "https://en.wikipedia.org/w/index.php?title=Zermelo–Fraenkel_set_theory&oldid=1312908109"
Categories:
Hidden categories:
[8]ページ先頭
©2009-2025 Movatter.jp