FOM: CH and second-order logic

William Tait wwtx at
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.

Bill Tait

More information about the FOM mailing list