[FOM] Platonism and Formalism

[Harvey Friedman]

Reply to Gaifman.

On 9/9/03 3:25 PM, "Haim Gaifman" <hg17 at columbia.edu> wrote:

>... But perhaps we
> should
> distinguished between describing one's *behavior* and giving an *account*
> of a certain view. For example, we believe that the system we employ is
> consistent. Can a formalist give grounds for this belief, over and above
> the brute fact that so far no contradiction has been discovered? Note that
> even
> the disjunction "S is consistent V S is inconsistent" involves an
> application of
> the excluded middle in an infinite domain, and it cannot be reduced to
> something more palatable to a finitist (along Hilbert's lines).
> 

I believe results will come in that will make everybody I know, from extreme
formalist to extreme Platonist, very uncomfortable.

A specific assertion first order assertion phi in the finite structure

(V(9),epsilon)

where V(9) is the 8th level of the cumulative hierarchy starting with V(0) =
emptyset, 

will be found with the following properties:

i) phi is simple and clear and natural;

ii) phi is generally regarded as a noteworthy mathematical truth if true;

iii) many close variants of phi are equally noteworthy and published
proofs/refutations of them by attractive standard methods appear, causing
there to be an attractive field of research surrounding statements like phi;

iv) phi and its close variants make just as good sense in V(omega) as they
do in V(9), with the situation in i),ii),iii) prevailing:

v) a proof of phi is published which is carried out within ZF + "there
exists a nontrivial elementary embedding from some V(kappa) into V(kappa),
where kappa is an inaccessible limit of inaccessible cardinals";

vi) a proof is also published of the following metamathematical result,
proved using only the basic properties of V(11) - i.e., the proof is
"carried out" within (V(11),epsilon), but itself appears as 100 pages in the
Annals of Mathematics.

*If phi is true then there is no proof of phi in the system ZF + "there
exists a nontrivial elementary embedding from some V(kappa) into V(kappa),
where kappa is an inaccessible cardinal", which is small enough to be an
element of V(7).* 

Some relevant background is as follows.

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;

3) if consistency is weakened to apply only to proofs which are elements of
V(7), then the same situation prevails.

4) it is well known that this system #) is substantially stronger than the
systems that the Platonistic wing of the set theory accept as consistent.

Beyond what I say I believe above, I believe that such a state of affairs
exists for any reasonably coherent set theoretic principle of any kind, even
ones that are far bolder than the existence of such elementary embeddings
under ZF. I.e., I believe that this is an unending phenomena that cannot be
shaken off in any way that has been suggested thus far.

Harvey Friedman