[FOM] First-order arithmetical truth

Francis Davey fjmd1 at yahoo.co.uk
Mon Oct 16 16:01:03 EDT 2006

Arnon Avron wrote:
> If you do not see or understand what are the (real) natural numbers, 
> then you do not see or understand any other concept which is
> define inductively. You dont understand what are the legal
> expression of any formal language, or what are the intended 
> proofs of a given formal system. So what can you understand
> and see? It seems to me that practically nothing. In fact,
> you cant (according to your own statements) even understand what
> Godel theorems are about! (they are about the intended structure
> of proofs in PA, but how can you tell this structure from another)

I suppose its because I have a computer science background that I tend 
to think of the formal system as coming first, and only being happy with 
things that can be expressed as formal systems.

Inductive definitions aren't a problem: they can be described quite well 
(as can all the mathematics I can think of) using formal methods that 
never need to rely on any notion of semantics.

NB: I am not denying either the usefulness of semantics (as a tool) nor 
any possible philosophical status to it. It may be that mathematics is 
"really" about objects and truths that can only be understood in terms 
of models, rather than in terms of deduction. I just don't know what 
those objects and truths are (perhaps yet).

However, it is not true to say that Godel's theorems (in context I 
assume his two incompleteness theorems) are "about the intended 
structure". The first incompleteness theorem is first described as a way 
of showing that certain statements of PM are formally undecidable. In 
other words they can be understood without any reference to a model of 
PM (or PA).


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