[FOM] First-order arithmetical truth/the intended model of arithmetic

[V.Sazonov@csc.liv.ac.uk]

Stephen Pollard wrote:

> The first-order number theoretic truths are exactly the first-order
> sentences in the language of arithmetic that follow from the axioms
> of Peano Arithmetic supplemented by the following version of the
> least number principle: "Among any numbers there is always a
> least." (This principle is not firstorderizable; but that doesn't
> make it unintelligible.)

Just vice versa! This principle only looks as self-evident as an 
extrapolation from some finite "pictures" of our real world. As to the 
above general form, I would consider it even meaningless (recall, e.g. 
the Berry paradox on "The smallest positive integer not definable in 
under eleven words".). Another thing is that there is a great 
temptation to postulate it (or Induction Axiom) under suitable 
formalisation. (Nowadays there is seemingly no other formalisation than 
first-order; what is called "second-order" is simply special version of 
two-sorted "first-order"). The reason of this temptation is not a 
"truth" of IA but rather the above extrapolation (a general scientific 
and methodological approach) and extraordinary applicability of IA in 
mathematics and (by transitivity only!) in the external world.


Quoting Francis Davey <fjmd1 at yahoo.co.uk> Fri, 13 Oct 2006:

> I'm worried by the idea that one can "see" that the Godel sentence is
> true. One obviously can't inside the model, but to do so outside looking
> in requires a more elaborate logical apparatus, which has its own problems.

I would rather say that one cannot "see" that Goedel sentence is true, 
but one can have a great temptation to postulate it as "true". Why such 
a temptation arises seems deserves to be explained in detail, probably 
using also psychological reasons and avoiding any mysticism and 
quasi-religious platonistic beliefs.

By the way, it can be criticised that the formal assertion Consis(PA) 
does adequately express the intuitive statement on consistecy of PA. 
Consis(PA) is only some strong extrapolation to the latter and is not 
automatically "true". And Goedel theorem just witnesses this fact.


Quoting Robert Black <Mongre at gmx.de> Fri, 13 Oct 2006:

> Francis Davey wrote:
>
>> . I might believe that the godel
>> sentence is true in the "intended model", but no-one has ever been able
>> to explain exactly what they mean by the intended model, so I am far
>> from sure about that. Maybe its a mystical ability I don't have or
>> haven't appreciated (*).
>> ...(*) in other words - how do I know there aren't a non-standard number of
> hadrons in the universe?
>
> It is indeed possible to doubt whether we can determinately pick out
> the intended model of arithmetic, i.e. to doubt whether we have a
> determinate understanding of the word 'finite' (for the intended
> model is the model in which each number only has finitely many
> predecessors), but then you're disabled from conjecturing over
> whether the number of hadrons in the universe might really be
> 'nonstandard' (i.e. infinite) since you won't understand
> 'nonstandard' either.

"Nonstandard" is in fact not a negation of "standard". It is quite 
"positive" concept, unlike the really fictitious "standard".

We are living in the "nonstandard", non-ideal world, and this can be 
easily observed. We also have some ideals which are usually 
non-observable.

I do not know as to hadrons, but I definitely have a clear intuition 
that ANY imaginary model of natural numbers (closed under successor and 
satisfying traditional axioms) is non-standard in the sense that it has 
(imaginably and even formalisably 
http://www.cs.nyu.edu/pipermail/fom/2006-February/009746.html, and in 
this sense observably) a strictly initial part of "feasible numbers" 
which is also closed under successor.

On the other hand, I absolutely do not understand what is the 
(allegedly unique) "intended", or "standard model", an "absolute ideal" 
(if it is not the *relative* concept formally defined in a framework 
like ZF and then, of course, non-questionable).


Vladimir Sazonov


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