FOM: Charles Silver on silly questions
John Mayberry
J.P.Mayberry at bristol.ac.uk
Mon Oct 12 09:40:21 EDT 1998
There is nothing wrong with the proof that all models of these
second order axioms are isomorphic (we don't have to talk about the
"standard structure" here). But for the proof to get off the ground we
must assume that there is a transfinite set with a power set. Of course
most mathematicians simply take that assumption for granted. But if you
don't take it for granted - and clearly Sazonov doesn't - then you are
left with the problem of how to lay the foundations for natural number
arithmetic. And in those circumstances Sazonov is right: the problem is
not that it is *untrue* that each natural number is obtained by
starting from 0 and iterating the successor function a finite number of
times; it is rather that it is not clear what such a claim means.
Dropping Dedekind's infinitary assumptions deprives us of his
mathematically precise analysis of the notion of finite iteration (his
theorem on definition by induction in Section 126 of his essay, and the
discussion in Section 130).
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John Mayberry
J.P.Mayberry at bristol.ac.uk
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