[FOM] A proof that ZFC has no any omega-models

meskew at math.uci.edu meskew at math.uci.edu
Mon Feb 18 17:54:29 EST 2013

> Solovay did not settle the question of whether ZFC proves the continuum
> does not have a total measure, he just showed that that is true iff ZFC
> disproves a measurable cardinal. And he did not need the existence of
> large cardinals to obtain this arithmetical result. Although not all
> arithmetical consequences of large cardinals follow from the consistency
> of the cardinals and don't require their actual existence, this particular
> arithmetical consequence   (absence of a ZFC disproof of a total measure
> on the continuum) requires only consistency of the measurable cardinal.

I know that his theorem does not establish in PA the statement Con(ZFC +
rvm), but it does establish it in the theory PA + Con(ZFC + measurable). 
In your previous post, you advocated the adoption of such assumptions as
axioms.  Thus by your standards, you should conclude not just the
conditional, "If Con(ZFC + measurable), then Con(ZFC + rvm)," but also the
consequent, and that Solovay settled the question.

> My more general point is that assuming models exist satisfying large
> cardinal axioms involves a much more concrete type of statement than
> assuming the cardinals themselves exist (since such models may be
> considered objects in second-order arithmetic), and therefore is a more
> palatable assumption to those mathematicians who question the definiteness
> of statements involving actual large cardinals. I am not claiming that
> this would change mathematical practice significantly, I am just saying
> that we should not assume more than we need to.

My point is that in your view you would have set theorists assuming ZFC +
T (where T is some natural extension such as a large cardinal assumption),
and from this proving a theorem P.  Then you would say thanks, but I'll
just accept the arithmetical corollary, "ZFC + T proves P," or some second
order arithmetical statement about models.  But you do not make any
recommendations for what kinds of T to use, and it seems you would just
let set theorists figure that out.  The point is that the real work is
done by set theorists who operationally assume ZFC + T.  Translating their
work into a scheme that boils down the arithmetical content seems to have
no mathematical impact, and indeed your prescription would be for set
theorists to go on doing what they are doing, and in particular to go on
assuming what they are currently assuming.  Hence my conclusion, the
advocacy of this scheme would be just a philosophical footnote, and should
not make set theory any more palatable to those who want more conservative
mathematical assumptions, unless their concerns are purely metaphysical.


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