[FOM] On Existence of Mathematical Objects

[Karlis Podnieks]

You may think of me as of a journalist who is trying to find out information
that may be important for students.

----- Original Message ----- 
From: "Harvey Friedman" <friedman at math.ohio-state.edu>
To: "fom" <fom at cs.nyu.edu>
Sent: Saturday, October 11, 2003 11:51 AM
Subject: Re: [FOM] On Existence of Mathematical Objects


> Reply to Sazonov.
>
> On 10/9/03 3:42 PM, "Vladimir Sazonov" <V.Sazonov at csc.liv.ac.uk> wrote:
>
> ...

> Perhaps Sazonov thinks that almost everybody else is talking
> *NOT about the natural number system*
> but rather talking about the
> **imaginary natural number system**.
> Then Sazonov may agree that the axiom scheme of induction for
> **imaginary natural numbers**
> is highly compelling, if not evident?
>
>...

For me, the induction principle is evident, but it says only that P(0) and
Ax(P(x)->P(x+1)) imply AxP(x). It seems, it does not say explicitly that
every natural number can be constructed by adding 1's to 0. Of course, we
can think that the induction principle says it implicitly. How is it
possible that the most fundamental mathematical structure allows only an
implicit description? How to get over this feeling? (We have a similar
feeling in set theory when trying to define finite sets.)

> ... The imaginary natural number system
> enjoys some very fundamental properties that the imaginary set system does
> not.
>

It seems so. I would wish a more detailed explanation of this for students.
Are these fundamental properties somehow impacted by the sad fact that every
such explanation involves "rules", i.e. a structure possibly equivalent to
natural numbers? Or, are these properties completely implied by this fact?
How to get over this feeling?

> Sazonov:
 > >May be PA will be once shown to be contradictory, too.
>
> Are you suggesting a fundamental defect in the imagination of working
> mathematicians?
>

I'm feeling baffled about the well known fact that "consistent universal
arithmetical statements are provable". For example, if, in some metatheory
M, Goldbach's Conjecture would be proved to be consistent with (a subset
of?) PA, then this conjecture would be proved in M as a theorem. (See my
exposition for students at http://www.ltn.lv/~podnieks/gt6a.html#s67). Is
this a "normal" situation? How to get over this feeling?

> Sazonov:
> > It would be disaster [inconsistency in Peano Arithmetic] only for those
who
> have superbeliefs.
> > Just as it was in the case with the old paradoxes in set theory.
>
> Wouldn't various theorems in real analysis and other areas of mathematics
> have to be rewritten and retaught with negations in front of them?
>

We could try to imagine this rewriting as somewhat similar to the already
existing "constructive rewriting" of the classical mathematics. I.e. the
result may be a more complicated and less "powerful" system...

>...
>
> Is every axiom of PA true in the imaginary natural number system?
>
> Harvey Friedman
>

Yes. Thank you for the term "the imaginary natural number system".

My questions are in no way intended as ironical objections. I'm simply
trying to use the biggest intuitions available on the FOM to answer
questions of some smartest students.

Best wishes,
Karlis.Podnieks at mii.lu.lv
www.ltn.lv/~podnieks
Institute of Mathematics and Computer Science
University of Latvia