Objection To The Axiom of Infinity
Further information: FinitismA common objection to Cantor's theory of infinite number involves the axiom of infinity. It is a generally recognized view by logicians that this axiom is not a logical truth. Indeed, as Mark Sainsbury has argued "there is room for doubt about whether it is a contingent truth, since it is an open question whether the universe is finite or infinite". Mayberry has noted that "The set-theoretical axioms that sustain modern mathematics are self-evident in differing degrees. One of them – indeed, the most important of them, namely Cantor's axiom, the so-called axiom of infinity – has scarcely any claim to self-evidence at all".
Another objection is that the use of infinite sets is not adequately justified by analogy to finite sets. Hermann Weyl wrote:
| “ | … classical logic was abstracted from the mathematics of finite sets and their subsets …. Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of set theory …." | ” |
Richard Arthur, philosopher and expert on Leibniz, has argued that Cantor's appeal to the idea of an actual infinite (formally captured by the axiom of infinity) is philosophically unjustified. Arthur argues that Leibniz' idea of a "syncategorematic" but actual infinity is philosophically more appealing. (See external link below for one of his papers).
The difficulty with finitism is to develop foundations of mathematics using finitist assumptions, that incorporates what everyone would reasonably regard as mathematics (for example, that includes real analysis).
Read more about this topic: Controversy Over Cantor's Theory
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