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

[Keith Brian Johnson]

I understand that the practice of mathematicians need not be altered by making their theorems just the conclusions of conditional proofs. But I don't think that failure to change the practice of mathematicians is really the point.
 
Suppose a group of people fabricated a very complicated notion of what a unicorn would be like, were one to exist, and said things like, "Unicorns are horned equines; unicorns have specialized internal organs; unicorns have intricate social structures," and so on. If they didn't preface their remarks by "If unicorns exist...," then it would be reasonable for a listener to inquire, "Do you really mean to imply that unicorns really exist?" Then the unicorn-theorists would have to say, "Oh, no--we're just reaching conclusions about what would be true were unicorns to really exist." 
 
Now, as long as everybody understands that the unicorn-theorists don't intend to make claims about actual unicorns, there's no pressing reason for them to reiterate the "If unicorns exist..." antecedent. They can just talk about unicorns as though they were talking about lions or tigers. But if not everybody understands that they do not intend to make claims about actual unicorns, then there's room for confusion.
 
Should mathematicians prove theorems that rely on certain unstated assumptions, and should not everyone understand that they are making those statements dependently upon those assumptions, then there will be room for confusion about what mathematicians think they are saying. Prefacing their remarks by "If these unstated assumptions are true..." just removes confusion. It seems to me to be a matter of clarity. 
 
But should those same mathematicians rely on those same unstated assumptions but not realize that they are doing so, then making those unstated assumptions explicit helps clarify not only those mathematicians' communications with other people but their own thought. While it isn't going to change how mathematicians do their work, the additional clarity of thought seems to me to be intrinsically important--if, that is, they really are relying on those unstated assumptions. And if they're not relying on those unstated assumptions, then it will be intrinsically important for them to know that they're not.
 
Keith Brian Johnson


>________________________________
> From: Monroe Eskew <meskew at math.uci.edu>
>To: Foundations of Mathematics <fom at cs.nyu.edu> 
>Sent: Monday, February 25, 2013 1:11 AM
>Subject: Re: [FOM] A proof that ZFC has no any omega-models
>  
>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
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