home *** CD-ROM | disk | FTP | other *** search
- Path: sparky!uunet!sun-barr!olivea!spool.mu.edu!sdd.hp.com!zaphod.mps.ohio-state.edu!think.com!news!columbus
- From: columbus@rachmaninoff.think.com (Michael Weiss)
- Newsgroups: sci.math
- Subject: Re: ZFC etc. (was Re: Report on Philosophies of Physicists)
- Message-ID: <COLUMBUS.92Sep15171406@rachmaninoff.think.com>
- Date: 16 Sep 92 00:14:06 GMT
- References: <716501145.10401@minster.york.ac.uk>
- <1992Sep15.184930.10080@guinness.idbsu.edu>
- Organization: Thinking Machines Corporation, Cambridge MA, USA
- Lines: 32
- NNTP-Posting-Host: rachmaninoff.think.com
- In-reply-to: holmes@opal.idbsu.edu's message of 15 Sep 92 18:49:30 GMT
-
- In article <1992Sep15.184930.10080@guinness.idbsu.edu>
- holmes@opal.idbsu.edu (Randall Holmes) answers the question:
- >
- >Are the proofs by Cohen and Godel are formal proofs? (I would think
- >so.) Can we identify the formal system in which the proofs were
- >performed?
-
- thus:
-
- They are rigorous arguments. No one ever does proofs in formal
- systems, in practice. The Godel proof is a construction in ZF (normal
- set theory without choice). The Cohen proof is a construction in ZFC
- (Choice is used, I believe).
-
- The relative consistency results of Godel and Cohen can be regarded as
- purely combinatorial statements about pushing symbols around,
- and as such can be expressed in the language of Peano arithmetic.
-
- In principle, the proofs of these relative consistency statements could be
- carried out in Peano arithmetic. Cohen discusses this briefly in the last
- chapter of his book "Set Theory and the Continuum Hypothesis".
- I believe the treatment in Shoenfield's book "Mathematical Logic" makes the
- same point.
-
- In practice, the proofs become far more intuitive if we adopt a less
- puritan attitude, and talk about models of ZF. Cohen makes free use of
- axiom SM (="there exists a model of ZF whose universe is a set and whose
- element-of relation is the standard element-of relation".) Cohen proves
- with ZF+V=L+SM that there is (for example) a model of ZFC + not CH.
- Naturally this implies the relative consistency result, Con(ZF) -->
- Con(ZFC+not CH). But if all you want is the relative consistency result,
- then a much weaker set of axioms (such as Peano arithmetic) will do.
-