> On 18 Jan 2019, at 09:49, Philip Thrift <cloudver...@gmail.com> wrote:
> 
> 
> The semantic view of theories and higher-order languages
> Laurenz Hudetz
> https://link.springer.com/article/10.1007/s11229-017-1502-0
> 
> "every family of set-theoretic structures has an associated language of 
> higher-order logic and an up to signature isomorphism unique model-theoretic 
> counterpart"
> 
> Several philosophers of science construe models of scientific theories as 
> set-theoretic structures. Some of them moreover claim that models should not 
> be construed as structures in the sense of model theory because the latter 
> are language-dependent. I argue that if we are ready to construe models as 
> set-theoretic structures (strict semantic view), we could equally well 
> construe them as model-theoretic structures of higher-order logic (liberal 
> semantic view). I show that every family of set-theoretic structures has an 
> associated language of higher-order logic and an up to signature isomorphism 
> unique model-theoretic counterpart, which is able to serve the same purposes. 
> This allows to carry over every syntactic criterion of equivalence for 
> theories in the sense of the liberal semantic view to theories in the sense 
> of the strict semantic view. Taken together, these results suggest that the 
> recent dispute about the semantic view and its relation to the syntactic view 
> can be resolved.

It cannot do that in the Mechanist Frame, where we cannot assume, neither a 
physical universe, nor analysis or set theory. Since recently, I have realised 
that we cannot even assume the induction axioms, but we can use it in the 
definition of the Löbian entities, whose existence is then a consequence of the 
theories without induction. Of course, we need induction at the meta level, and 
even the whole of informal mathematics, like in any science pointing toward 
some reality independent of us.

Bruno





> 
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