What’s Wrong with Infinite Quantifiers?

Abstract

The foundation stones of platonist mathematics are the infinite quantifiers, ∀ and ∃, regarded straightforwardly as infinite conjunction and infinite disjunction. For example, Goldbach’s conjecture (that every even number greater than 2 is the sum of two primes) is understood as an infinite conjunction ‘4 is the sum of two primes and 6 is the sum of two primes and 8 is the sum of two primes and…’. Wright (1980, I, §7) quotes this example and asks, since we understand the decidable predicate ‘is the sum of two primes’, the binary operation ‘and’, and the sequence of even numbers (which we can effectively enumerate), how can we possibly doubt that Goldbach’s conjecture is a meaningful proposition? The answer is obvious: the dubious part is the three dots ‘…’. I shall spend this chapter examining these strange punctuation marks. They can be understood in two ways, traditionally called ‘potential infinity’ and ‘actual infinity’. ‘Actual infinity’ means viewing the quantifiers as infinitary analogues of conjunction and disjunction: I shall argue that this is meaningless. ‘Potential infinity’ means viewing the quantifiers, and all other apparent references to infinities, in terms of the counting algorithm and other computational procedures: I shall defend this basic notion of computation against Wittgenstein.