FOM: second-order logic is a sterile myth

[Pat Hayes]

Steve--

Since the topic seems to be so important to you, I wonder if you would tell
us how you reconcile yourself to the fact that every consistent first-order
theory (including, one hopes, ZFC) has a countable model?

There seems to be a central snag with the idea of choosing any formal
language as the foundation for mathematics, which is that formal logics
with finite expressions and R.E. theorem sets tend to be compact, and hence
can be consistently interpreted within a countable universe. Now, of
course, one is always free to claim that such a restricted class of
interpretations isn't what one has in mind, while still using the language
consistently. One can take the position that when one quantifies over all
sets, one means indeed to quantify over all sets; and the fact that the
formal truth-conditions cannot enforce this intention is irrelevant to what
one intends to express when using the language. To repeat, I respect such
an intellectual position (in fact I think it is more or less forced on us
by the metamathematical facts.)

However, I can see no coherent justification for taking this position with
respect to first-order logic, while passionately refusing to allow anyone
to take it with respect to second-order logic. If you take the first-order
quantifiers of ZFC to really range over uncountable structures, then why do
you get so hot under the collar about other people insisting that
second-order quantification really does range over the domain of all
relations? Neither position is warranted by a semantic theory of the formal
language in question which can yield a completeness result for any proof
system.

Pat Hayes

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