[FOM] Could PA be inconsistent?

Karlis Podnieks Karlis.Podnieks at mii.lu.lv
Tue May 18 01:40:37 EDT 2004

Let us recall a fragment from an interview given by Harvey Friedman to a
"journalist" (full text at

>> ... The imaginary natural number system
>> enjoys some very fundamental properties that the imaginary set system
>> not.
> It seems so. I would wish a more detailed explanation of this for
> Are these fundamental properties somehow impacted by the sad fact that
> 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.

End of quote.

How to explain this "privileged" position of natural numbers among other
mathematical structures? Couldn't it be caused by the "sad fact" that any
formalization of any mathematical structure involves some formal syntax,
i.e. - something very similar to a (or, "the"?) natural number system?
If we are trying to formalize structures by means of formal syntax, then the
structures that are equivalent to this formal syntax, should look "better"
than other structures?

Doesn't this phenomenon re-occur in the model theory of non-countable

Best wishes,
Karlis.Podnieks at mii.lu.lv
Institute of Mathematics and Computer Science
University of Latvia

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