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

[Monroe Eskew]

On Feb 21, 2013, at 7:35 PM, "Timothy Y. Chow" <tchow at alum.mit.edu> wrote:
> 
> It seems to me that the camp of people whose "concerns are purely metaphysical" is pretty large.  Isn't metaphysics one of the main driving forces behind conservative mathematical assumptions?
> 
> Whether or not it's "just a philosophical footnote," it seems like a significant observation to me if it adequately addresses the metaphysical objections to set theory.
> 
> Tim



Please explain how someone who accepts all arithmetical consequences of large cardinals is "conservative."  This viewpoint is perhaps the most liberal possible with respect to arithmetical propositions.

Let me try to paint the picture again.  Suppose those with the metaphysical worries gain control of the government.  There are some set theorists proving theorems.  Some metaphysically worried government agent comes along and says, "Your theory seems to make too bold ontological assumptions.  I would be much more comfortable if you deflated it all and only claimed in the end the arithmetical consequences of your work."  Then the good citizens of the nation decide that they will put the appropriate prefixes and suffixes to their work so that only the arithmetical consequences are officially claimed.  The government is satisfied, but except for the addenda, it is the same work.

To me this picture looks like nothing has really changed.  Some people with philosophical worries forced the mathematicians to put little ornaments around their work, but mathematical practice did not actually change.  The mathematicians went on to assume the same assumptions, A, and all they had to do to show good citizenship was to first say, "Suppose A," and at the end say, "Therefore A implies...."  The government didn't actually change how mathematics is done.  So what was their supposed conservative vision all about?

Monroe