[FOM] Ruling out Nonstandard Models of first-order PA

Leon Horsten Leon.Horsten at hiw.kuleuven.be
Wed Oct 25 03:19:52 EDT 2006

Dear Robert,

You ask:

>  Are there other such informal ideas?
>     There is a possible third tantalizingly suggested by,-

>  The Tennenbaum-Kreisel Theorem...
> The tantalizing suggestion is that, formal PA supplemented by the    
> informal but clear and distinct idea,
> [III] the arithmetical "+" is reckonable, or calculable.

Volker Halbach and I have tried to work out this suggestion in a paper
of ours:

[2005] Halbach,  V. & Horsten, L. Computational structuralism.
Philosophia Mathematica 13.2(2005), p. 174-186.

The tricky part, it seems to us, is to avoid a circularity charge.
Namely, an opponent could object that the concept of calculability
presupposes the notion of natural number. So we formulated our
explication of what first-order arithmetic is about in terms of a
concept of reckonability or calculability which does not presuppose
the notion of natural number. We took the calculability involved to be
on notation systems instead of on the numbers themselves. I.e., we
essentially made use of another idea that you mentioned:

> [1.a.iii] still less abstractly,--such an idea of the Hindu-Arabic    
> decimal system of numerals 1, 2, ...., 9, 10, 11, ... (ideally    
> continuing ad infinitum)

Best regards,
Leon Horsten

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