FOM: What is the standard model for PA?

Vladimir Sazonov sazonov at
Mon Mar 16 13:19:24 EST 1998

Torkel Franzen wrote:
>    Validimir Sazonov says:
>    >On the other hand it is not me who asserts determinacy or
>    >clarity about the notion "natural number". It seems that my
>    >position is more safe: If somebody asserts this, let him explain
>    >at least what does it mean.
>   "Determinate", like "real", is what J.L.Austin called a "trouser
> word": it's the the opposite (or a range of possible opposites) that
> wears the trousers. I can only say that I don't see anything
> indeterminate or unclear about the notion "natural number", and ask
> you to point out to me where you see an unclarity or indeterminacy.

You don't see anything indeterminate or unclear about the notion 
"natural number" or you see that and how it is determinate? 

It looks as if you present a theorem and, instead of giving the proof, 
ask the opponent whether he sees a counterexample. I realize that this 
discussion is not about a theorem. But I only say that I do not 
understand a thesis while you seems have asserted that you undedrstand 
it. Anyway, I am glad to reply. 

>   In your comments, you remark that "and so on" doesn't even say
> that there is no last natural number. Sure, but you're not saying
> that there is any unclarity about whether or not there is a
> last natural number. You know there isn't, just as I know it.
> So no unclarity on that score.

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. 

> You also point out that "and so on"
> might cover only x for which log log x < 10. Sure, but you're quite
> as aware as anybody else that on the ordinary understanding of
> "natural number", log log x < 10 doesn't hold for every x. So there's
> no unclarity or indeterminacy there either.

Yes, I do know that it does not hold according to such and such axioms.  
Bu how can I know what else should hold in the hypothetical "standard 
model"? Which numbers larger than 2^1000, etc. will occur in this
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. I have some 
illusion of understanding. But I realize that this is illusion which has 
very unclear ground and try to be careful. Let me also note, that I do 
not understand the meaning of quantifiers over such unclear model and 
also why Induction Axiom is true for it. But I am ready to postulate IA 
and to discuss (in a positive manner) why it is reasonable to postulate 
it, to reduce it to transfinite induction up to epsilon-0, etc., etc.  
All of this is very reasonable, interesting and can be understood in 
some way without reference to the "true" standard model. Of course, we 
can go further and further. I only do not know how much further.  This 
is indeterminate. 

>   I don't think it's necessary to demonstrate the mathematical and
> applied usefulness of a proposed arithmetical law in this context, but
> only to give some explanation of why one should believe it to be
> true.

No absolute truth and no beliefs, except those based on experiments
with logarithm function. It is true for *feasible* numbers represented
as real finite sets of physical objects (pebbles, and the like).

> The notion of "feasible number" may be a very useful and natural
> one to invoke in many contexts and applications. But this affords no
> grounds whatever for the claim that the ordinary notion of natural
> number is in any way unclear or indeterminate,

Actually I wrote that "this low seems to demonstrate (or illustrate) 
that there is actually no hope on having a unique notion of natural 
numbers and of the absolute "truth" even in arithmetic". I can assert 
nothing definite about the notion of "standard model" which I do not 
understand (except what is provable from the axioms which are usually 
postulated for this model).  I can only assert more carefully about some 
kind of illustration of why or how this notion is indeterminate. It is 
indeterminate somewhat as the notion of feasible numbers. We begin to 
count and say "and so on". In the case of feasible numbers we say 
frankly (following A.S.Esenin-Volpin) "and so on, up to exhaustion".  In 
the case of "the" standard model, I even do not know what do we say.  
"And so on" without any comments sais almost nothing.  What I am sure, 
is that I do not understand this notion and I never seen satisfactory 
explanation even of what does it mean. Only formal approach via ZF or 
noncritical view helps us to get illusion that everything is OK with 
"this" standard model.  I have nothing against illusions. I would like 
only to be able to recognize what is an illusion and what is not. Then I 
can decide what to do with it. Do you agree that "standard model" is an 
illusion created by our imagination? Can illusion be completely 
determinate? Why do you need this "completely determinate illusion"?  
What will you do with it? As to me, I would like to switch on or off my 
illusions depending on the situation. It is hardly possible to do this 
if we cannot recognize them. 

> or that experience
> should prompt us to adopt "for all x, log log x <10" as an
> "arithmetical law".

It is arithmetical in the sense that it is about numbers. What do YOU
by "arithmetical"?

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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