[FOM] Comment on Church's Thesis (Harvey Friedman)

[Hartley Slater]

Alexander Zenkin writes:
>	Wilfried Hodges writes in one place: " . . .the author observes quite
>correctly that the PROOF of Cantor's theorem (on the uncountability of the
>real line) depends on acceptance of actual infinity."
>
>	How is it possible that not simply mathematical theorem 
>(Cantor's theorem
>on the uncountability of continuum), not "the intuitive meaning of the
>theorem", but its MATHEMATICAL PROOF may depend on an acceptance of the
>"vague, intuitive, philosophical" notion of "actual infinity" "what is
><certainly> not a mathematical theorem or axiom"?

If you assume there is a countable infinite set, then the set of its 
subsets will have a larger cardinality, but without that assumption 
there is no proof that anything has that larger cardinality. Thus 
Cantorian Set Theories have to include an axiom of infinity, although 
most mathematicians working on the subject leave it to others, 
usually philosophers, to question whether the axiom is true.  Hodges 
points out that it is needed mathematically, Wittgenstein, for 
instance, has disputed whether it is true - see, for instance, Victor 
Rodych's 'Wittgenstein's Critique of Set Theory' The Southern Journal 
of Philosophy, 2000.
-- 
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, M207 School of Humanities
University of Western Australia
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