FOM: What is the standard model for PA?

Vladimir Sazonov sazonov at
Wed Mar 18 12:21:44 EST 1998

Torkel Franzen wrote:
>  Validimir Sazonov says:
>    >You don't see anything indeterminate or unclear about the notion
>    >"natural number" or you see that and how it is determinate?
>   I don't see anything indeterminate or unclear about the notion
> "natural number". "Seeing that it is determinate", as far as I can
> determine, amounts only to not seeing anything indeterminate or
> unclear about it. This doesn't exclude the possibility that you might
> convince me that there is something indeterminate or unclear about
> it.

Probably this corresponds to what I described as an illusion of 
the standard model. Of course I also have some such illusion.  
(You probably will not agree to call it this way.) As any 
illusion it has some objective roots (on which we probably can 
agree). It is interesting to discuss such roots, to understand 
how they work in the creation of this illusion. For example, we 
know some geometrical illusions arising from some simple figures 
on a sheet of paper (stairs or the like). Who considers 
corresponding images in our mind as real?

>    >No, I do not know. More precisely, I know that I may consider both
>    >alternative, each being reasonable in its own way. Let us fix that
>    >there is NO last natural number because this is usually *postulated*
>    >for "the standard model". This still does not fix this standard
>    >model, just how *long* is it.
>   "The standard model" is jargon. You knew everything that I know about
> the natural numbers (from the present philosophical point of view) long
> before you had heard of "the standard model". One of the things you
> learned was that there isn't any largest natural number - given any number
> k, there is a larger one, k+1. It's not clear to me what you consider
> "unfixed".

>From the point of view of the above discussed illusion, when 
we concentrate *only* on it, we will hardly see anything 
unfixed.  It comes in mind the "full" induction axiom on 
"arbitrary" properties of natural numbers which would fix the 
standard model completely (up to isomorphism) as in the 
framework of set theory. You know that the term "arbitrary" is 
too problematic here. It bothers me too much to consider this 
model as "really" fixed.

Also I would like to have the following picture of "creating" 
the natural numbers which seems do not contradict to the 
traditional view, but is of somewhat different flavour.  We 
count 0,1,2, etc. We iterate successor operation to get + and 
iterate + to get * and then exponential, etc. We realize that 
the more is "operative" ability of considered operations, the 
"longer" is the row of arising in this way natural numbers. We 
consider these operations as abstract, non-realistic ones.  We 
introduce them only in the hope to overcome the currently 
considered infinity.  E.g. if we have only addition operation 
(as *total* recursive function) then multiplication is only 
partial recursive and 2^1000 will hardly arise in this row, even 
if we consider it as infinite (no last number). Then we 
postulate that * is also total and get 2^1000 reachable with the 
help of multiplication by two.  However, we will never consider 
2^1000 as reachable by the direct (feasible) iteration of the 
successor operation. Thus, we have many different infinite 
natural number series. I do not see how we will get "all" 
natural numbers in this process even potentially. Also I do not 
understand what does it mean this "all". All the attention is 
concentrated here on fuzziness of all these intermediate 
infinities and in the most degree of the "whole" process. The 
simplest one is the first infinity corresponding to feasible 

>   >Yes, I do know that it [for all x, log log x < 10] does not hold
>   >according to such and such axioms.
>   Axioms are secondary. People who have never heard of the Peano
> axioms and wouldn't be interested in hearing about them learn and
> apparently accept that log is an unbounded function, on the basis of
> their informal understanding of "natural number". Is there some
> confusion or unclarity in this?

I believe that the main feature of mathematics is *provability* 
of their results. I cannot imagine proofs without some kind of 
proof rules and axioms. We learn mathematics even at school by 
some training repeated again and again how to do something 
*correctly* (of course, together with corresponding intuitions).  
That is why we are able to recognize correct proof from 
incorrect one (even without knowing, say what is the name of the 
rule modus ponens, reductio ad absurdum, etc.; however, e.g. my 
students, even not the best ones, actually know after school 
even the name of the latter rule). In particular we implicitly 
learned at school some (idea of) induction or iteration rule, i.e., 
essentially Peano Arithmetic, which allows us to *deduce* easily 
that exponential (= iterated multiplication) is total function 
and logarithm is bounded.

>   >If you say "all", I do not understand what does this mean. It is unclear
>   >what is "all" EVEN in the more reliable case of feasible numbers which
>   >have a physical, perceiving meaning in our real world.
>   Then we're back to the beginning. All the natural numbers are 0,s(0),s(s(0))
> and so on. What's unclear or indeterminate about this?
>   As for the "feasible numbers", I don't know what you mean by "the
> more reliable case". The notion of "feasible number" is very unclear
> and indeterminate - isn't that so?

This notion is *comparatively* clearer that an illusory idea of 
standard model. Who do not know what is *physically* written 
string in a finite alphabet? It is less clear about f.n. how to 
work with them in a formal axiomatic system like PA. However, 
how many peoples did try to do this?  On the other hand, there 
were some attempts which show that this is possible. I believe 
that only (let, say, hypothetical) possibility of formalization 
of feasibility concept gives us some hope on including it in 
mathematics as a "first-class citizen".

>    >I have some llusion of understanding.
>   Well, I would say that you rather have an illusion of not
> understanding.  Or, in less confrontational terms, I would say that
> you have certain ideas about what is required to understand something,
> or what is required for something to make determinate sense. I am
> certainly not in a position to say that these ideas are wrong, that
> you are mistaken, that your pursuits are misguided. On the contrary, I
> think it's a good thing that people pursue such ideas and
> inclinations, including stuff like Yesenin-Volpin's consistency proof
> for ZFC (or NF). Where I think you are plain wrong is only in
> presenting your ideas and inclinations as though they revealed some
> defect in standard practice.

It is rather not in standard practice but around it. I would not 
like to assert anything global. But I really feel rather strong 
discomfort when I see some assertions on "absolute truth" (not in 
the technical sense of model theory) and the like. 

Vladimir Sazonov
Program Systems Institute,  	| Tel. +7-08535-98945 (Inst.),
Russian Acad. of Sci.		| Fax. +7-08535-20566
Pereslavl-Zalessky,		| e-mail: sazonov at
152140, RUSSIA			|

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