FOM: Re: CH and 2nd-order validity
kanovei at wmwap1.math.uni-wuppertal.de
Mon Oct 16 12:06:22 EDT 2000
> From owner-fom at math.psu.edu Mon Oct 16 17:18:04 2000
> From: "Roger Bishop Jones" <rbjones at rbjones.com>
> Subject: FOM: Re: CH and 2nd-order validity
> Does CH have a truth value under the semantics of first order set theory?
> I expect the semantics of first order set theory to give the truth
> conditions for all sentences of first order set theory, including CH.
CH has the truth value, defined as follows:
if there is a 1-1 correspondence between countable ordinals and
reals then CH is true, otherwise it is false.
This is the only (modulo modifications) mathematical definition
what CH is and what does it mean that it is "true", which is just
one and the same. And this is well known since Cantor.
It is also well known since Goedel and Cohen that there is no
mathematically sound way to actually determine whether CH is true
or false (in the abovementioned sense, which is the only reasonable one).
Semantically, this can be expressed as follows:
(if there is at least one model of ZF then)
there are models of ZFC where CH is true
and there are models of ZFC where CH is false.
Philosophers should dislike this, of course.
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