FOM: Charles Silver on silly questions

[John Mayberry]

	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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