[FOM] Platonism and Formalism

[Harvey Friedman]

Reply to Solovay.

Friedman wrote: 

>> 1) there apparently is no mathematical logician willing to assert that they
>> are convinced that
>> 
>> #) ZF + "there exists a nontrivial elementary embedding from some V(kappa)
>> into V(kappa), where kappa is an inaccessible cardinal"
>> 
>> is consistent;
>> 
>> 2) nor do any think that there are good grounds for believing it. Nor have
>> any put forward plans for gathering evidence about it;
>> 
> 

Solovay wrote:

> Harvey,
> 
> Have you asked Hugh Woodin what he thinks about # ? Back when I
> was doing set-theory we talked about principles which might well be much
> stronger than even the variant of # where "inacessible" is replaced by
> "inaccessible limit of inaccessibles". I admit that I never asked him if
> he believed the principles in question were consistent. Of course, they
> certainly aren't **true** by well-known results of Kunen. {So I presume
> the "it" that is discussed in 2) is the consistency of # rather than #
> itself.}
> 

Yes, of course, "it" refers to the consistency of #).

I asked Woodin by email, and he says by email to me that he does not have an
opinion on whether or not #) is consistent. He says that it is a very
interesting question.

Judging by the lack of "structure theory", I would surmise that Woodin, as
well as many other set theorists, might say similar things about proof
theoretically weaker hypotheses such as the existence of a nontrivial
elementary embedding from a rank + 1 into itself.

E.g., it would be interested to see what kind of opinion John Steel has
about consistency of such things, including #).

Let me say a little more about the point of my posting of 9/10/03 8:58AM.

The conjecture is that any issue that naturally arises concerning the
"extent" of the outer reaches of the set theoretic universe - whatever that
may be - has an equally natural corresponding issue in the first few levels
of the finite cumulative hierarchy, that is very closely related. So closely
related that progress on the latter issue does not seem possible without
progress on the former issue. The relationship between these two can of
course be stated in much more technical terms, as I did in my posting of
9/10/03 8:58AM. 

Harvey Friedman