FOM: Re: theory-edge mailing list, tautologies

[Martin Davis]

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