infinity in set theory, the property of a set whereby it has a proper subset whose members can be placed in one-to-one correspondence with all the members of the set, as the even integers can be so arranged in respect to the natural numbers by the function f(x) = x/2, namely: Devised by Richard Dedekind in defiance of the age-old intuition that no part of a thing can be as large as the thing, this set-theoretical definition of ‘infinity’, having been much acclaimed by philosophers like Russell as a model of conceptual analysis that philosophers were urged to emulate, can elucidate the putative infinity of space, time, and even God, his power, wisdom, etc. If a set’s being denumerable – i.e., capable of having its members placed in one-to-one correspondence with the natural numbers – can well appear to define much more simply what the infinity of an infinite set is, Cantor exhibited the real numbers (as expressed by unending decimal expansions) as a counterexample, showing them to be indenumerable by means of his famous diagonal argument. Suppose all the real numbers between 0 and 1 are placed in one-to-one correspondence with the natural numbers, thus: Going down the principal diagonal, we can construct a new real number, e.g., .954 . . . , not found in the infinite ‘square array.’ The most important result in set theory, Cantor’s theorem, is denied its full force by the maverick followers of Skolem, who appeal to the fact that, though the real numbers constructible in any standard axiomatic system will be indenumerable relative to the resources of the system, they can be seen to be denumerable when viewed from outside it. Refusing to accept the absolute indenumerability of any set, the Skolemites, in relativizing the notion to some system, provide one further instance of the allure of relativism.
More radical still are the nominalists who, rejecting all abstract entities and sets in particular, might be supposed to have no use for Cantor’s theorem. Not so. Assume with Democritus that there are infinitely many of his atoms, made of adamant. Corresponding to each infinite subset of these atoms will be their mereological sum or ‘fusion,’ namely a certain quantity of adamant. Concrete entities acceptable to the nominalist, these quantities can be readily shown to be indenumerable. Whether Cantor’s still higher infinities beyond F1 admit of any such nominalistic realization remains a largely unexplored area. Aleph-zero or F0 being taken to be the transfinite number of the natural numbers, there are then F1 real numbers (assuming the continuum hypothesis), while the power set of the reals has F2 members, and the power set of that F3 members, etc. In general, K2 will be said to have a greater number (finite or transfinite) of members than K1 provided the members of K1 can be put in one-to-one correspondence with some proper subset of K2 but not vice versa.
Skepticism regarding the higher infinities can trickle down even to F0, and if both Aristotle and Kant, the former in his critique of Zeno’s paradoxes, the latter in his treatment of cosmological antinomies, reject any actual, i.e. completed, infinite, in our time Dummett’s return to verificationism, as associated with the mathematical intuitionism of Brouwer, poses the keenest challenge. Recognition-transcendent sentences like ‘The total number of stars is infinite’ are charged with violating the intersubjective conditions required for a speaker of a language to manifest a grasp of their meaning.
See also CONTINUUM PROBLEM , SET THE- OR. J.A.B.