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From: play@mcvax.UUCP (Andries Brouwer)
Newsgroups: net.math
Subject: Re: Re: Nova's Mathematical Mystery Tour (truth of CH)
Message-ID: <529@mcvax.UUCP>
Date: Tue, 12-Mar-85 12:21:56 EST
Article-I.D.: mcvax.529
Posted: Tue Mar 12 12:21:56 1985
Date-Received: Sun, 17-Mar-85 21:03:31 EST
References: <143@ihlpa.UUCP> <460@petsd.UUCP> <6353@boring.UUCP> <350@talcott.UUCP>
Reply-To: play@mcvax.UUCP (Andries Brouwer)
Organization: CWI, Amsterdam
Lines: 35

In article <350@talcott.UUCP> gjk@talcott.UUCP (Greg Kuperberg) writes:
>> The same must apply to classical mathematicians.  Even though CH is in-
>> dependent of the axioms of ZF Set Theory, it is conceivable (although high-
>> ly implausible) that someone will some day come up with new methods of
>> mathematical reasoning that are *obviously* valid, using which CH can be
>> decided.  The situation is different from the one concerning Euclid's Fifth
>> Postulate.  None of Euclid's axioms is `true', obvious or not.  There are
>> `geometries' in which there can be several lines through two given points
>> (e.g., great circles on a sphere).  However, a `mathematics' in which both
>> a proposition and its negation can be true is unacceptable to classicists,
>> intuitionists and constructivists alike.
>... 
>>      Lambert Meertens
>
>Really?  I had always thought that there were two kinds of universe:
>in one class, the CH is true, and in the other it is false.  That is, just
>like there are other geometries in which Euclid's fifth is false, there are
>"universes" in which CH is true and other "universes" in which it is false.
>If this duality is valid, how can one possibly come up with new methods by
>which CH can be decided?  Or is this duality valid in the first place?
>---
>			Greg Kuperberg

In a mathematical theory you use axioms and inference rules to arrive
at theorems. The axioms are usually made explicit; they are not considered
obvious, but are assumed as a starting point. The rules of inference are
seldom made explicit - it is assumed that mathematicians can recognise
valid reasoning. What Lambert says (I think) is that it is conceivable
(but unlikely) that somebody comes up with an obviously valid inference
rule that would enable us to decide CH from the ZFC axioms.
I do not agree, since the current inference rules already allow you to
construct models in which CH holds and models in which it doesnt hold.
Stronger inference rules deciding CH would thus lead to a contradiction.
Something that is conceivable to me is that one might come up with new
axioms for Set Theory that are *obviously* valid and would decide CH.