[FOM] On Existence of Mathematical Objects

[Harvey Friedman]

Reply to Podnieks.

On 10/15/03 1:57 AM, "Karlis Podnieks" <Karlis.Podnieks at mii.lu.lv> wrote:

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

I'll do anything for students (smile).

Friedman 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?
>> 
>> ...

Podnieks wrote:
> 
> 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.

But how is that to be formulated? The usual view is that it can be
formulated only by the induction principle, with P viewed as ranging over
"all" unary predicates, or "all" unary predicates defined on the natural
numbers, or by some other idea that also introduces a primitive notion
beyond that of natural number and successor (or any specific operations on
natural numbers).

This principle is already very powerful even if there is no explicit idea of
"all" unary predicates (on the natural numbers) available. It is NOT
powerful unless one has something like, in addition to successor, addition
and multiplication.

When no idea of "all" unary predicates is acknowledged, the principle is
normally viewed as an "approximation", whose strength is according to what
existence principles one has for unary predicates (of natural numbers).

>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.)

How would it be possible that the most fundamental mathematical structure(s)
allow explicit description? It would SEEM have to be in terms of something
yet more fundamental, contradicting "most fundamental".

>> ... 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?

For students: 

The natural numbers under successor has no proper substructure.

Note that

The set theoretic universe under membership has a proper substructure.

E.g., remove the empty set.

However, one is naturally lead to considering statements like this.

1. The set theoretic universe has a proper elementary substructure.
2. The set theoretic universe has a proper elementary substructure that does
not form a set. 
3. The set theoretic universe has a proper second order elementary
substructure.
4. The set theoretic universe has a proper second order elementary
substructure that does not form a set.
5. The set theoretic universe has two proper second order elementary
substructures, neither of which is a subset of the other.

For students: analyze the status of 1-5 as statements in class theory.

>> 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?

Your claim 

"consistent universal arithmetic statements are provable"

is false. The correct claim is

"consistent universal arithmetic statements are true"

and as a Corollary,

"provably consistent universal arithmetic statements are provable".

>> Is every axiom of PA true in the imaginary natural number system?
> 
> Yes. Thank you for the term "the imaginary natural number system".
> 

Let's see what Sazonov has to say about this terminology.

Harvey Friedman