FOM: conservative extensions of nominalistic theories

hartry field hf18 at
Mon Jan 31 19:16:19 EST 2000

Ketland writes 
>Field’s 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). 
>Field’s 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 doesn’t work! Adding set theory to certain synthetic 
>descriptions of spacetime is non-conservative. This is closely 
>connected to Godel’s 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
be weakened 
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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