FOM: second-order logic is a myth

Stephen G Simpson simpson at
Mon Mar 8 20:04:42 EST 1999

Robert Black 4 Mar 1999 11:18:06 writes:
 > OR You insist that plural quantification is just a sneaky way of talking
 > about sets.  (Despite the joky way in which I presented it, this is
 > actually my own preferred option, and it appears to be Steve's too.

Yes, that is my preferred option, at least in those cases when plural
quantification can't be straightforwardly paraphrased in terms of
singular quantification.  Your Napoleon example is one such case.  (By
the way, this example seems a little stilted.  I have never heard
anyone except an academic philosopher talk this way.  Do you have an
example that is less artificial?)

 > ... But then it seems that you will have to concede that set
 > theoretical reasoning pervades our thought concerning just about
 > any topic,

I don't concede that.  Besides, there is a distinction between sets in
the real world (sets of cards on a table, for instance) and sets in
the sense of ZF.  It may be that real-world sets play a role in much
everyday reasoning, but ZF-sets include a lot of higher infinities
that have no obvious real-world counterpart.  What is the real-world
counterpart of aleph_omega?  The connection of aleph_omega to
real-world sets seems very tenuous.

 > and by the topic-neutrality criterion set theory will be part of
 > logic.

I certainly don't concede that.

My point is that set theory is not part of logic.  To make this
absolutely clear, we would probably need to go into the history of set
theory.  We would need to discuss how set theorists arrived at the
current understanding of ZF et al as first-order theories, with
logical axioms (i.e. the common axioms and rules of predicate
calculus) and non-logical axioms (i.e. the specifically set-theoretic

I think you more-or-less conceded the point when you said:

 > I agree that set theory is normally studied in first-order logic,
 > and rightly so.  (How else could one study the universe of sets
 > except by examining consequences of axioms while using an
 > underlying logic in which a correct proof is effectively
 > recognizable as such?)

I would rephrase this as follows: The only way to systematically study
any subject is to develop a theory based on logic together with axioms
about the specific subject matter.  This applies even when the subject
matter is sets.

-- Steve

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