From: holmes@opal.idbsu.edu (Randall Holmes)
Newsgroups: sci.physics,sci.math
Subject: Re: Report on Philosophies of Physicists
Date: 14 Sep 92 19:12:07 GMT
Organization: Boise State University Math Dept.

In article <1992Sep13.174721.23818@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>In article <TORKEL.92Sep13095337@bast.sics.se> torkel@sics.se (Torkel Franzen) writes:
>>In article <1992Sep13.050206.18067@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.
>>Stanford.EDU (Vaughan R. Pratt) writes:
>>
>>   >Let # denote the Goedel coding of "ZFC is inconsistent."  What I
>>   >interpret you to be asking is, what reality R could possibly be
>>   >described by ZFC+# under the assumption that ZFC is consistent in R?
>>   >Right?
>>
>>  No, nothing like that. I am just wondering what anybody would mean by
>>saying that ZFC+# can be used as a foundation for mathematics. A foundation
>>for what? In what sense a foundation?
>
>Well, I thought I'd answered just that, but maybe I was jumping the
>gun.  The meaning of ZFC+# as a foundation for mathematics is no
>different from say ZFC+V=L (all sets constructible), or any other
>addition to ZFC.  It just means the same mathematics you've always
>done, plus anything new you can get from this additional postulate.
>
>You might object that # has no mathematical content.  That remains to
>be seen.  The test for this is not the form of # but whether
>interesting new mathematics results.  If # "goes nowhere" as they say
>in foundations (I'll call this "fruitful"), ZFC+# will have proved to
>be essentially ZFC, just the old foundations encumbered with some
>pointless baggage (note that # is a lot of baggage, encoding not only
>ZFC itself but all of proof mechanics).  But if it opens up a hot new
>field with exciting results not obtainable in ZFC then your position
>that # has no mathematical content becomes hard to defend.  (I owe this
>insight to Sol Feferman, who expressed a mild feeling that # would go
>nowhere, but that if it did yield interesting mathematics it would
>reopen the question of the appropriateness of ZFC.  I would add to that
>prospect the likelihood that people would then start looking for a nice
>(at least shorter) rephrasing of # abstracting its real "engine".)
>
>To make this yet more concrete, Harvey Friedman (and there may be
>others I don't know about) has for a long time been investigating the
>mathematical content of not-#, to see whether it is equivalent to a
>nice statement (which would make # similarly nice since the class of
>nice statements, whatever that is, is presumably closed under
>negation).  He says that thus far he hasn't found anything really
>convincing for ZFC, though he has nice stuff off to the side, at ZC+V=L
>(Zermelo with Choice and Constructibility but without Fraenkel's
>Replacement).
>
>This seemingly quixotic quest is put in perspective by the celebrated
>Paris-Harrington result, that the infinite Ramsey theorem is
>independent of Peano Arithmetic.  This was proved by showing that it
>was equivalent to (a slight strengthening of) not-# for PA, the
>consistency of Peano Arithmetic.  The corresponding situation for ZFC
>is that, while various nice propositions are known to be independent of
>(i.e. undecided in) ZFC, none of these proofs to date have been via the
>route that succeeded for Paris-Harrington in PA.  If a nice restatement
>of not-# exists then one immediately obtains that this nice statement
>is independent of ZFC, *via the independence of inconsistency* a la
>Paris-Harrington.

This is flat wrong.  # is false in the standard model of arithmetic
(in ZFC, if you like); the Paris-Harrington Theorem is true there.
Thus the PH theorem cannot be a strengthening of #.


>
>It is probably wishful thinking to hope that one of not-# or # might
>turn out to imply one of CH or not-CH (continuum hypothesis).
>
>(Note in passing: relative to ZFC, we have now seen three independent
>types of propositions, undecided, fruitful, and nice.  Well, almost
>independent, if we restrict "fruitful" to undecided propositions.  The
>nice class is closed under negation but the fruitful class is surely
>not, I would even expect this to be easy to prove for a suitably
>general class of notions of "fruitful".)
>
>-- 
>======================================================| God found the positive
>Vaughan Pratt   pratt@cs.Stanford.EDU   415-494-2545  | integers, zero was
>======================================================| there when He arrived.


-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu
Newsgroups: sci.physics