FOM: CH and second-order logic
wwtx at midway.uchicago.edu
Tue Oct 17 15:25:23 EDT 2000
John Steel (Oct 16) wrote
> I believe he suffers here from an illusion that lurks in
>a lot of philosophical discussions on mathematics. This is the idea
>that it is the function of formal semantics--1st or 2nd order model
>theory--to assign meaning to our mathematical language. Formal semantics
>models certain aspects of meaning assignments, but if one uses it to
>communicate the meaning of language L, one is essentially translating L
>into the language of set theory. Obviously, one cannot communicate, or
>sharpen, the meaning of the language of set theory this way.
I agree completely that it is an illusion and that it lurks. But it
is also true that the sets R(alpha) in any two standard models of
second-order set theory containing alpha are isomorphic. [Zermelo
1930] So, if CH holds in one, it holds in the other. I suppose that
this is a semantic argument(?)---for what it is worth.
Kanovei (Oct 16) wrote concerning the independence of CH
>Philosophers should dislike this, of course.
On the contrary, not all philosophers have a taste for clear skies
and desert landscapes (Quine, somewhere); some of us like it messy.
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