Zermelo–Fraenkel set theory
From Wikipedia, the free encyclopedia
|
Zermelo–Fraenkel set theory, with the axiom of choice, commonly abbreviated ZFC, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics.
IntroductionZFC consists of a single primitive ontological notion, that of set, and a single ontological assumption, namely that all individuals in the universe of discourse (i.e., all mathematical objects) are sets. There is a single primitive binary relation, set membership; that set a is a member of set b is written a Failed to parse (Missing texvc executable; please see math/README to configure.): \in b (usually read "a is an element of b" or "a is in b"). ZFC is a one-sorted first-order theory; hence the background logic is first-order logic. These axioms govern how sets behave and interact. In 1908, Ernst Zermelo proposed the first axiomatic set theory, Zermelo set theory. This axiomatic theory did not allow the construction of the ordinal numbers; while most of "ordinary mathematics" can be developed without ever using ordinals, ordinals are an essential tool in most set-theoretic investigations. Moreover, one of Zermelo's axioms invoked a concept, that of a "definite" property, whose operational meaning was not clear. In 1922, Abraham Fraenkel and Thoralf Skolem independently proposed operationalizing a "definite" property as one that could be formulated in first-order logic, with all atomic formulas involving set membership or identity. From their work emerged the axiom schema of replacement. Appending this axiom, as well as the axiom of regularity, to Zermelo set theory yields the theory denoted by ZF. Adding the axiom of choice (AC) to ZF yields ZFC. When a mathematical result requires the axiom of choice, this fact is often stated explicitly. The reason for singling out AC in this manner is that AC is inherently nonconstructive; it posits the existence of a set (the choice set), without specifying just how that set is to be constructed. Hence results proved using AC may involve sets that, although they can be proved to exist (at least if one is not committed to a constructivist ontology), can never be constructed explicitly. For instance, the axiom of choice implies that any set can be well-ordered. While we cannot construct a well-order for the set of real numbers R, AC guarantees that such an order exists. ZFC has an infinite number of axioms because the Replacement axiom is actually an axiom schema. Montague (1961) showed that the set theories ZFC and ZF cannot be axiomatized by a finite set of axioms. On the other hand, Von Neumann–Bernays–Gödel set theory (NBG) can be finitely axiomatized. The ontology of NBG includes classes as well as sets; a set is a class that is a member of another class. NBG and ZFC are equivalent set theories in the sense that any theorem about sets (i.e., not mentioning classes in any way) which can be proved in one theory can be proved in the other. Because of Gödel's second incompleteness theorem, the consistency 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 there being a weakly inaccessible cardinal, something whose existence is not provable in ZFC (unless ZFC is inconsistent). Nevertheless, it is unlikely that ZFC harbors an unsuspected contradiction; if ZFC were inconsistent, it is widely believed that that fact would have been uncovered by now. This much is certain — ZFC is immune to the classic paradoxes of naive set theory: Russell's paradox, the Burali-Forti paradox, and Cantor's paradox. Drawbacks of ZFC that have been discussed in the literature include:
AxiomsThere are many equivalent formulations of the ZFC axioms; for a rich but somewhat dated discussion of this fact, see Fraenkel et al (1973). The following particular axiom set is that of Kunen (1980). English descriptions have been added for clarity.
The converse of this axiom is a consequence of the substitution property of equality.
3) Axiom schema of specification (also called the Axiom schema of separation or the Axiom schema of restricted comprehension): If z is a set, and Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\! is any property which may characterize the elements x of z, then there is a subset y of z containing those x in z which satisfy the property. The restriction to z is necessary to avoid Russell's paradox and its variants. More formally, let Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\! be any formula in the language of ZFC, and suppose that the free variables appearing in Failed to parse (Missing texvc executable; please see math/README to configure.): \phi\! are among Failed to parse (Missing texvc executable; please see math/README to configure.): x,z,w_1,\ldots,w_n\! . Then:
Specification is part of Z but redundant in ZF, because there are variants of ZF's axiom schema of replacement that turn Specification into a theorem.
Pairing is part of Z but is redundant in ZF, because ZF's axiom schema of replacement (in combination with the axiom of infinity) turns Pairing into a theorem.
there is a set A containing every set that is a member of some member of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{F}
.
6) Axiom schema of replacement: Informally, if the domain of a function f is a set, then the range of f is also a set, subject to a restriction to avoid paradoxes. Formally, let the formula Failed to parse (Missing texvc executable; please see math/README to configure.): \phi \! and its free variables Failed to parse (Missing texvc executable; please see math/README to configure.): x,y,A,w_1,\ldots,w_n \! be as described in the axiom schema of specification. Then:
Here the quantifer Failed to parse (Missing texvc executable; please see math/README to configure.): \exists ! y means that only one Failed to parse (Missing texvc executable; please see math/README to configure.): y\! exists, up to equality.
is some set. From pairing, union, specification, and extensionality above, one can show that Failed to parse (Missing texvc executable; please see math/README to configure.): S(x)\! exists uniquely and that the empty set exists uniquely, if any set Failed to parse (Missing texvc executable; please see math/README to configure.): x \! exists. Together with the axiom of infinity which asserts the existence of a set, the latter fact makes redundant Kunen's axiom (not shown) asserting the existence of at least one set. 7) Axiom of infinity: There exists a set X such that the empty set Failed to parse (Missing texvc executable; please see math/README to configure.): \varnothing is a member of X and whenever y is in X, so is S(y).
8) Axiom of power set: For any set x there is a set y that contains every subset of x.
Here Failed to parse (Missing texvc executable; please see math/README to configure.): z \subseteq x is an abbreviation for Failed to parse (Missing texvc executable; please see math/README to configure.): \forall q (q \in z \Rightarrow q \in x) .
Alternative forms of axioms 1-8 are often encountered. For example, the axiom of pairing (#4) is often changed to say that for any sets x and y there is a set containing exactly x and y. Similarly, the axioms of union, replacement, and power set are often written to say that the desired set contains only those sets which it must contain. An axiom is sometimes added which asserts that the empty set exists. For an example of some of these variations, see the list of axioms in Jech [2003]. The Axiom of choice has many equivalent statements; that is, there are many statements that can be proved equivalent to axiom 9 using axioms 1-8. The name "axiom of choice" refers to one such statement, namely that there exists a choice function for every set of nonempty sets. Since the existence of a choice function for finite sets is easily proved, this formulation is interesting because of what it asserts about certain infinite sets. The list above includes two infinite axiom schemes. In his 1957 PhD thesis, Montague proved that, if ZFC is consistent, no finite axiomatization of ZFC is possible. See also
Bibliography
External links
cs:Zermelo-Fraenkelova teorie množin da:Zermelo-Fraenkels aksiomer de:Zermelo-Fraenkel-Mengenlehre es:Axiomas de Zermelo-Fraenkel fr:ZFC ko:체르멜로-프란켈 집합론 it:Teoria degli insiemi di Zermelo - Fraenkel pl:Aksjomaty Zermelo-Fraenkela pt:Axiomas de Zermelo-Fraenkel sr:Зермело-Френкел теорија скупова sv:ZFC tr:Zermelo-Freankel küme kuramı uk:Теорія множин Цермело-Френкеля |
