نظرية فئات زرملو-فرنكل

نظرية مجموعات زيرميلو-فرانكل مع بديهية الاختيار (الإنكليزية: , Zermelo–Fraenkel set theory with the axiom of choice)، والتي يشار إليها اختصاراً ZFC، هي أحد النظم البديهية التي طرحت في بداية القرن العشرين لتشكيل نظرية مجموعات دون التناقضات التي عانت منها نظرية المجموعات البسيطة مثل متناقضة رسل.

وبشكل أكثر تحديداً، إن نظرية ZFC خالية من الفهم غير المقيد.

تعد نظرية مجموعات زيرميلو-فرانكل اليوم نظرية المجموعات البديهية المعيارية، وتعد من أشهر أسس الرياضيات الحالية.

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 البديهيات

All formulations of ZFC imply that at least one set exists. Kunen includes an axiom, in addition to the following, which directly asserts the existence of a set. Many authors require a nonempty domain of discourse as part of the semantics of the first-order logic in which ZFC is formalized. The axiom of infinity (below) also asserts that at least one set exists, as it begins with an existential quantifier.

1. Axiom of extensionality: Two sets are equal (are the same set) if they have the same elements.

$\forall x\forall y[\forall z(z\in x\Leftrightarrow z\in y)\Rightarrow x=y].$

The converse of this axiom follows from the substitution property of equality. If the background logic does not include equality "=", x=y may be defined as an abbreviation for the following formula (Hatcher 1982, p. 138, def. 1):

$\forall z[z\in x\Leftrightarrow z\in y]\land \forall z[x\in z\Leftrightarrow y\in z].$

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

$\forall x\forall y[\forall z(z\in x\Leftrightarrow z\in y)\Rightarrow \forall z(x\in z\Leftrightarrow y\in z)],$

which says that if x and y have the same elements, then they belong to the same sets (Fraenkel et al. 1973).

2. Axiom of regularity (also called the Axiom of foundation): Every non-empty set x contains a member y such that x and y are disjoint sets.

$\forall x[\exists a(a\in x)\Rightarrow \exists y(y\in x\land \lnot \exists z(z\in y\land z\in x))].$

3. Axiom schema of specification (also called the axiom schema of separation or of restricted comprehension): If z is a set, and $\phi \,$ 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 $\phi \,$ be any formula in the language of ZFC with free variables among $x,z,w_{1},\ldots ,w_{n}\,$. So y is not free in $\phi \,$. Then:

$\forall z\forall w_{1}\ldots w_{n}\exists y\forall x[x\in y\Leftrightarrow (x\in z\land \phi )].$

This axiom is part of Z, but can be redundant in ZF, in that it may follow from the axiom schema of replacement, with (as here) or without the axiom of the empty set.

The axiom of specification can be used to prove the existence of the empty set, denoted $\varnothing$, once the existence of at least one set is established (see above). A common way to do this is to use an instance of specification for a property which all sets do not have. For example, if w is a set which already exists, the empty set can be constructed as

$\varnothing =\{u\in w\mid (u\in u)\land \lnot (u\in u)\}$.

If the background logic includes equality, it is also possible to define the empty set as

$\varnothing =\{u\in w\mid \lnot (u=u)\}$.

Thus the axiom of the empty set is implied by the nine axioms presented here. The axiom of extensionality implies the empty set is unique, if it exists. It is common to make a definitional extension that adds the symbol $\varnothing$ to the language of ZFC.

4. Axiom of pairing: If x and y are sets, then there exists a set which contains x and y as elements.

$\forall x\forall y\exists z(x\in z\land y\in z).$

This axiom is part of Z, but is redundant in ZF because it follows from the axiom schema of replacement applied to any two-member set. The existence of such a set is assured by either the axiom of infinity, or by the axiom of the power set applied twice to the empty set.

5. Axiom of union: For any set ${\mathcal {F}}$ there is a set A containing every set that is a member of some member of ${\mathcal {F}}.$

$\forall {\mathcal {F}}\,\exists A\,\forall Y\,\forall x(x\in Y\land Y\in {\mathcal {F}}\Rightarrow x\in A).$

6. Axiom schema of replacement: Let $\phi \,$ be any formula in the language of ZFC whose free variables are among $x,y,A,w_{1},\ldots ,w_{n}\,$, so that in particular B is not free in $\phi \,$. Then:

$\forall A\,\forall w_{1},\ldots ,w_{n}[(\forall x\in A\exists !y\phi )\Rightarrow \exists B\forall x\in A\exists y\in B\phi ].$

Less formally, this axiom states that if the domain of a definable function f is a set, and f(x) is a set for any x in that domain, then the range of f is a subclass of a set, subject to a restriction needed to avoid paradoxes. The form stated here, in which B may be larger than strictly necessary, is sometimes called the axiom schema of collection.

7. Axiom of infinity: Let $S(x)\,$ abbreviate $x\cup \{x\}\,$, where $x\,$ is some set. Then there exists a set X such that the empty set $\varnothing$ is a member of X and, whenever a set y is a member of X, then $S(y)\,$ is also a member of X.

$\exists X\left[\varnothing \in X\land \forall y(y\in X\Rightarrow S(y)\in X)\right].$

More colloquially, there exists a set X having infinitely many members. The minimal set X satisfying the axiom of infinity is the von Neumann ordinal ω.

8. Axiom of power set: Let $z\subseteq x$ abbreviate $\forall q(q\in z\Rightarrow q\in x).$ For any set x, there is a set y which is a superset of the power set of x. The power set of x is the class whose members are all of the subsets of x.

$\forall x\exists y\forall z[z\subseteq x\Rightarrow z\in y].$

Alternative forms of axioms 1-8 are often encountered, some of which are listed in Jech (2003). Some ZF axiomatizations include an axiom asserting that the empty set exists. The axioms of pairing, union, replacement, and power set are often stated so that the members of the set x whose existence is being asserted are just those sets which the axiom asserts x must contain.

9. Well-ordering theorem: For any set X, there is a binary relation R which well-orders X. This means R is a linear order on X such that every nonempty subset of X has a member which is minimal under R.

$\forall X\exists R(R\;{\mbox{well-orders}}\;X).$

Given axioms 1-8, there are many statements provably equivalent to axiom 9, the best known of which is the axiom of choice (AC), which goes as follows. Let X be a set whose members are all non-empty. Then there exists a function f, called a "choice function," whose domain is X, and whose range is a set, called the "choice set," each member of which is a single member of each member of X. Since the existence of a choice function when X is a finite set is easily proved from axioms 1-8, AC only matters for certain infinite sets. AC is characterized as nonconstructive because it asserts the existence of a choice set but says nothing about how the choice set is to be "constructed." Much research has sought to characterize the definability (or lack thereof) of certain sets whose existence AC asserts.

 انظر أيضاً

• أسس الرياضيات
• Inner model
• Large cardinal axiom
• Non-well-founded set theory

Related axiomatic set theories:

• Morse-Kelley set theory
• Von Neumann-Bernays-Gödel set theory
• Tarski-Grothendieck set theory
• Constructive set theory
• Internal set theory

 مراجع

• Alexander Abian, 1965. The Theory of Sets and Transfinite Arithmetic. W B Saunders.
• -------- and LaMacchia, Samuel, 1978, "On the Consistency and Independence of Some Set-Theoretical Axioms," Notre Dame Journal of Formal Logic 19: 155-58.
• Keith Devlin, 1996 (1984). The Joy of Sets. Springer.
• Abraham Fraenkel, Yehoshua Bar-Hillel, and Azriel Levy, 1973 (1958). Foundations of Set Theory. North Holland. Fraenkel's final word on ZF and ZFC.
• Hatcher, William, 1982 (1968). The Logical Foundations of Mathematics. Pergamon.
• Thomas Jech, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.
• Kenneth Kunen, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.
• Richard Montague, 1961, "Semantic closure and non-finite axiomatizability" in Infinistic Methods. London: Pergamon: 45-69.
• Patrick Suppes, 1972 (1960). Axiomatic Set Theory. Dover reprint. Perhaps the best exposition of ZFC before the independence of AC and the Continuum hypothesis, and the emergence of large cardinals. Includes many theorems.
• Gaisi Takeuti and Zaring, W M, 1971. Introduction to Axiomatic Set Theory. Springer Verlag.
• Alfred Tarski, 1939, "On well-ordered subsets of any set,", Fundamenta Mathematicae 32: 176-83.
• Tiles, Mary, 2004 (1989). The Philosophy of Set Theory. Dover reprint. Weak on metatheory; the author is not a mathematician.
• Tourlakis, George, 2003. Lectures in Logic and Set Theory, Vol. 2. Cambridge Univ. Press.
• Jean van Heijenoort, 1967. From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931. Harvard Univ. Press. Includes annotated English translations of the classic articles by Zermelo, Fraenkel, and Skolem bearing on ZFC.
• Zermelo, Ernst (1908), "Untersuchungen über die Grundlagen der Mengenlehre I", Mathematische Annalen 65: 261–281, doi:10.1007/BF01449999  English translation in *Heijenoort, Jean van (1967), "Investigations in the foundations of set theory", From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, Source Books in the History of the Sciences, Harvard Univ. Press, pp. 199–215, ISBN 978-0674324497 
• Zermelo, Ernst (1930), "Über Grenzzablen und Mengenbereiche", Fundamenta Mathematicae 16: 29–47, ISSN 0016-2736, http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=16 

 روابط خارجية

• Stanford Encyclopedia of Philosophy articles by Thomas Jech:
  • Set Theory;
  • Axioms of Zermelo–Fraenkel Set Theory.
• Metamath version of the ZFC axioms — A concise and nonredundant axiomatization. The background first order logic is defined especially to facilitate machine verification of proofs.
  • A derivation in Metamath of a version of the separation schema from a version of the replacement schema.
• Eric W. Weisstein, Zermelo-Fraenkel Set Theory at MathWorld.