FOM: conservative extensions of nominalistic theories
hf18 at is4.nyu.edu
Mon Jan 31 19:16:19 EST 2000
>Fields 2 main claims are that
> (i) we can always eliminate mathematics from any physical
> theory to obtain a nominalistic theory (i.e., no quantifiers
> ranging over numbers, or sets).
> (ii) if we add mathematics to a nominalistic theory, we always
> get a conservative extension.
>He substantiated these two claims by, indeed, giving a nominalistic
>version of Newtonian gravitational physics (including a nominalistic
>or synthetic treatment of spacetime) and by arguing that any model
>of a nominalistic theory N can be expanded to a model of the result
>of adding ZFC to N (if this were true, then N+ZFC would have to be
>a conservative extension of N).
>Fields idea is that adding
>mathematics (i.e., set theory) would just be useful, but
>dispensable, instrument for finding things out about the concrete
>HOWEVER. Damn!!! It doesnt work! Adding set theory to certain synthetic
>descriptions of spacetime is non-conservative. This is closely
>connected to Godels theorems.
It depends what you mean by 'adding'. If you don't allow set-theoretic
vocabulary into the comprehension axioms used in the synthetic physical
theory then the extension IS conservative. And that's all my program really
required, as I
argued in "On Conservativeness and Incompleteness" JP 1985, a reply to the
Shapiro paper that Ketland cites.
(Admittedly, the representation theorems used in the original book need to
slightly if we don't expand the schemas by adding set-theoretic vocabulary;
but the weakened versions, described in the paper just mentioned,
seem adequate for all practical purposes.)
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