FOM: Re: theory-edge mailing list, tautologies
Martin Davis
martin at eipye.com
Tue Jul 31 14:08:53 EDT 2001
At 04:01 PM 7/31/2001 +0100, Roger Bishop Jones wrote:
>In response to <JoeShipman at aol.com> Tuesday, July 31, 2001 4:45 AM
>
>| The Continuum Hypothesis or its negation cannot have anything to say about
>| ... any ... mathematical statements
>| which can be formulated arithmetically.
>
>Can you give a brief explanation for the non-specialist of why this is the
>case?
Suppose A is an arithmetic statement and that either CH --> A or -CH --> A
is provable from the Zermelo-Fraenkel axioms (ZF). In the model of ZF found
by G\"odel in which CH is true, each sentence in the language of set theory
is interpreted relative to that model. BUT ARITHMETIC SENTENCES ARE
ABSOLUTE - MEANING THAT THEIR INTERPRETATION IN THE MODEL IS JUST THE
STATEMENT ITSELF. So from CH --> A and modus ponens one gets A, relative to
the model, and therefore, A itself. For -CH --> A, similar considerations
apply to the models Cohen found in which -CH is ture.
Martin
Martin Davis
Visiting Scholar UC Berkeley
Professor Emeritus, NYU
martin at eipye.com
(Add 1 and get 0)
http://www.eipye.com
More information about the FOM
mailing list