FOM: Re: CH and 2nd-order validity
John Steel
steel at math.berkeley.edu
Mon Oct 16 11:46:46 EDT 2000
Roger Bishop Jones wrote:
> Does CH have a truth value under the semantics of first order set
theory?
> If that semantics is exclusively determined by the semantics of first
order
> logic and the axioms of first order set theory (as Hilbert might have
> insisted) then it does not.
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.
The meaning of the language of set theory is determined by its use, in
mathematics, science, and everyday life. One learns its meaning by using
it, just as one learns one's mother tongue. In its use, the language of
set theory is neither 1st order nor second order; those are properties of
certain formal mathematical models of possible meaning assignments to its
syntax. (I believe there was some discussion a while back about whether we
do or should use a 1st or 2nd order language. I have no idea what the
difference would be.)
The mistaken idea the meaning of the language of set theory is
determined by assigning a model to it lies behind a lot of the (excuse my
French) baloney written on Skolem's paradox.
Harvey hit the nail on the head when he said:
ZF is not about models of ZF. ZF is about sets.
John Steel
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