Axiomatic Theory of Sets

the formulation of set theory as a formal (axiomatic) system.

The primary initiating stimulus for the construction of the axiomatic theory of sets was the discovery of paradoxes (anomalies), that is, contradictions, inG. Cantor’s “naive” theory of sets which had been proposed as a foundation of classic mathematics. All these paradoxes (for example, Cantor’s paradox connected with the examination of the “set of all sets,” or Russell’s paradox, which considers “the set of all sets which do not include themselves as elements”) are based on the unlimited application in Cantor’s set theory of the principle of induction (or abstraction), according to which there exists for every property a set which consists of all objects endowed with this property; this principle, actually, is included in the first statement of all classical expositions of set theory: “We shall consider arbitrary sets of elements of an arbitrary nature,” and so on.

The first of the known systems of the axiomatic theory of sets is the Zermelo-Fraenkel, or ZF, system formulated in 1908 by E. Zermelo and amplified in 1921–22 and later by A. Fraenkel. In the ZF system, the principle of induction is replaced by some of its special cases: the axiom of the existence of a pair (x,y) of any (given) sets x and y; the axiom of the existence of the union of all the elements of an arbitrary set x into a new set S(x); the axiom of the existence of set P(x) of all the parts of an arbitrary set x; the axiom of the existence of an infinite set and the so-called axiomatic schemes of segregation (according to which for every set x and property φ, there exists a set of elements x that possess the property φ) and substitution (which confirms that for every one-to-one mapping of elements of the set xdescribed in the language of the ZF system, there exists a set z into which the elements of the set ξ can be mapped). Not included under the principle of induction is the so-called axiom of choice, which deals with the existence of “the set of representatives”—that is, the set which contains exactly one element from each nonempty, pairwise nonintersecting set. As in every other system of axiomatic theory of sets, the ZF system also postulates the axiom of extensionality, according to which sets consisting of the same elements coincide. Sometimes certain other axioms with more specific significance are included in the ZF system. The formulas of the ZF system are derived from the “elementary formulas” of the form x ∊ y (“x belongs to y”) by means of the predicate calculation.

Many modifications of the ZF system subsequently appeared, as did systems which differed from the ZF in that “poor” aggregates of elements (which led to paradoxes) were not entirely excluded from consideraton but were admitted as “proper classes”—that is, sets that could not exist as elements of other sets; this idea was put forward by G. Neuman and later developed by P. Bernays, a Swiss mathematician, K. Gödel, and others. These systems, unlike the ZF, can be stated by means of finite numbers of axioms.

Another approach to the axiomatic theory of sets was the theory of types developed by B. Russell and A. N. Whitehead (in England, 1910–13) and its various modifications in which the applications of the axiom of induction have no limitations typical of the ZF and other systems, but the very language of the theory is altered. Instead of a single alphabet of variables x, y, z, . . . , an infinite sequence of alphabets x_l, y _l, z ₁, . . . ; x₂, y₂, z₂, . . . ; x_n, y_n, Z _n,....;... of various “types” η is introduced, and the elementary formulas are of the form x_nEЄy_n+1 orx_n = y_n. The theory of types is constructed on the basis of the predicate calculation with various forms of variables (which, upon the natural replacement of x_nEЄy_n+1 by y_n+1 ((x_n)) and x_n = = y_n by xⁿ~y_n may be considered as systems of expanded predicate calculation and not of set theory). In the NF (new foundation) system introduced by the American mathematician W. O. Quine in 1937, both approaches are combined: the language of the NF is the same as that of the ZF, but the axioms of induction must be derived from the axioms of the theory of types by removing the subscripts on the variables.

The problem of (relative) consistency of various systems of axiomatic theory of sets and their separate axioms has been studied. In 1940, Gödel demonstrated the relative consistency of the axiom of choice and the continuum hypothesis for the Σ system and the ZF which he described; ultimately this result was obtained in the theory of types (the weakest of the systems which we have described) and then in the NF system (in corresponding form). In 1963 the American mathematician P. J. Cohen demonstrated for ZF and X as well the relative consistency of negation of the continuum hypothesis, including the case in which the axiom of choice is included in the ZF system. He showed, further, that the ZF system can include, without creating a contradiction, the axiom which asserts that the continuum cannot be well-ordered; the negation of the axiom of choice follows immediately from this axiom.

The above-mentioned limitations upon the induction principle or the language of the system are sufficient to prevent any of the known paradoxes from arising in the axiomatic theory of sets. However, the problem of absolute consistency requires the incorporation of essentially new ideas, in view of Gödel’s incompleteness theorem. Specifically, the appearance of the proof of the consistency of the ZF (and the theory of types, excluding NF) in 1960, required the application of the means of the so-called ultraintuitionism.