Intuitionism - Intuitionism and Infinity

Intuitionism and Infinity

Among the different formulations of intuitionism, there are several different positions on the meaning and reality of infinity.

The term potential infinity refers to a mathematical procedure in which there is an unending series of steps. After each step has been completed, there is always another step to be performed. For example, consider the process of counting: 1, 2, 3, …

The term actual infinity refers to a completed mathematical object which contains an infinite number of elements. An example is the set of natural numbers, N = {1, 2, …}.

In Cantor's formulation of set theory, there are many different infinite sets, some of which are larger than others. For example, the set of all real numbers R is larger than N, because any procedure that you attempt to use to put the natural numbers into one-to-one correspondence with the real numbers will always fail: there will always be an infinite number of real numbers "left over". Any infinite set that can be placed in one-to-one correspondence with the natural numbers is said to be "countable" or "denumerable". Infinite sets larger than this are said to be "uncountable".

Cantor's set theory led to the axiomatic system of ZFC, now the most common foundation of modern mathematics. Intuitionism was created, in part, as a reaction to Cantor's set theory.

Modern constructive set theory does include the axiom of infinity from Zermelo-Fraenkel set theory (or a revised version of this axiom), and includes the set N of natural numbers. Most modern constructive mathematicians accept the reality of countably infinite sets (however, see Alexander Esenin-Volpin for a counter-example).

Brouwer rejected the concept of actual infinity, but admitted the idea of potential infinity.

"According to Weyl 1946, 'Brouwer made it clear, as I think beyond any doubt, that there is no evidence supporting the belief in the existential character of the totality of all natural numbers ... the sequence of numbers which grows beyond any stage already reached by passing to the next number, is a manifold of possibilities open towards infinity; it remains forever in the status of creation, but is not a closed realm of things existing in themselves. That we blindly converted one into the other is the true source of our difficulties, including the antinomies – a source of more fundamental nature than Russell's vicious circle principle indicated. Brouwer opened our eyes and made us see how far classical mathematics, nourished by a belief in the 'absolute' that transcends all human possibilities of realization, goes beyond such statements as can claim real meaning and truth founded on evidence." (Kleene (1952): Introduction to Metamathematics, p. 48-49)

Finitism is an extreme version of Intuitionism that rejects the idea of potential infinity. According to Finitism, a mathematical object does not exist unless it can be constructed from the natural numbers in a finite number of steps.