FOM: What is the standard model for PA?

[Vladimir Sazonov]

Dear Torkrel,

You wrote in particular 

>   The line of thought that you summarize as leading to different
> infinite natural number series is one that may well be worth
> pursuing  (although I admit that I don't myself put much faith in
> Yesenin-Volpin's consistency proof for ZFC). "All numbers" does
> indeed become an indeterminate notion on this line of thought.

It seems we have reached some understanding one another, at 
least each one's "line of thought". (Also note, that I did not 
say anything on Yesenin-Volpin's consistency proof for ZFC.)

>   >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.
>
>   I quite agree that the notion of "arbitrary property of natural
> numbers" is indeterminate, and I don't at all think that the full
> induction axiom for arbitrary properties of natural numbers can be
> invoked to establish the determinateness of the natural numbers.
> This does not imply that there is anything indeterminate or unclear
> about the natural numbers, or the quantifier "for every natural
> number".

I see from you reply that this indeterminateness of arbitrary 
property of natural numbers bothers you somewhat differently 
than me. Of course to make conclusions like mine there should be 
some additional impulse. It may be related with feasible 
numbers. It seems that the picture of generating "all" the 
natural numbers from feasible numbers by repeated (actually also 
feasible) iteration of arithmetical operation is also implicit 
in our intuitive understanding of this notion. Either we 
deliberately disregard it *according* to the ordinary 
mathematical education, or not. The second alternative requires 
some additional work. 

I think that after realizing that the natural numbers *may be 
seen* as constituting a very indeterminate "set" it is difficult 
to return to older, I would say, oversimplified picture as if 
nothing was happened. At least this is my case.  Probably you 
are able to be so "solid" in your opinion to not change your 
belief (or what it is) in standard model and simultaneously to 
realize possible vagueness of the same(?) model. 

By the way, do you see now "indeterminateness of arbitrary 
property" of natural numbers in rather short segment 
0,1,2,...,1000 of natural numbers as in the case of "all" 
numbers or this set is still completely determinate for you? 

>    >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.
>
>   Yes, but the *justification* for these arguments is not found in
> axioms, but in our understanding of the concepts involved.

Our understanding and "justification" goes *in terms*, and via 
numerous repetitions of using some formal rules. Otherwise, how 
to teach children to mathematics? (I believe that this is 
applied not only to children.) These rules are not so explicitly 
formulated as in the predicate calculus or the like, but they 
are still sufficiently determinate to follow them. All of this 
may look like the *pure* "understanding of the concepts 
involved" just because rules are *hidden*.

For example, when you are bicycling do you realize completely 
which way to use handlebars to preserve vertical position? Even 
if you know corresponding rule ("turn in the direction of 
slope"), does it help you very much to learn bicycling? Do you 
realize that now you are actually using this rule and you did 
learn exactly this rule by training when you first learned 
bicycling?

Also,
> I believe that for very many people it's a lot easier to recognize
> that addition is commutative than it is to recognize an inductive
> proof of the commutativity of addition as a proof.

Take it as axiom which has experimental confirmation.

>   >Who do not know what is *physically* written
>   >string in a finite alphabet?
>
>   The notion of a physically written string is full of uncertainties
> and indefiniteness. What is required for an arrangement of chalk dust,
> pencil lead particles, and so on, to constitute a written string in
> a particular finite alphabet?

Of course I agree. There is both determinateness of each clearly 
written string on a reliable surface and uncertainty of the 
"set" of "all" feasible strings. See also my previous posting in 
reply to Simpson.


Best wishes,
Vladimir Sazonov
--
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