FOM: Second order logic
John Mayberry
J.P.Mayberry at bristol.ac.uk
Tue Mar 9 07:26:31 EST 1999
In his reply to Robert Black, Steve Simpson draws the distinction
between "sets in the real world" and "sets in the sense of ZF". The
example of the former that he gives is a set of playing cards. The
point of the example must be that the playing cards are "in the real
world". But what about the set, S, that they compose? Is that "in the
real world"? If not, why not? But if S is in the real world, what about
S union {S}, is that in the real world? Where do we draw line here? Of
course these are not pure sets. But then in Zermelo's own formulation
of his system, he allowed "urelements" to occur in his sets, so S would
be a "set in the sense of ZF" for Zermelo himself. What about the
fundamental sequence (to use Zermelo's terminology) composed of finite
sets generated from the set S in the manner suggested (i.e. just like
the von Neumann natural numbers but starting with S instead of the
empty set)? Is that set in the real world? (Maybe the question ought to
be just: is that a set?). To take the Zermelo-Fraenkel system seriously
you have to ask these questions. And, of course, your answers may be
unfavourable to that system.
He goes on to say "The only way to systematically study any
subject is to develop a theory based on logic together with axioms
about the specific subject matter". Preumably the logic he has in mind
here is conventional 1st order logic. But if you are going to advocate
a procrustean policy of this sort, surely you have to give some sort of
argument to show that your system of logic is up to the job. And how
are you going to do that without using set theory of some sort? The
completeness of the logical axioms and rules is the critical question
here.
John Mayberry
School of Mathematics
University of Bristol
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John Mayberry
J.P.Mayberry at bristol.ac.uk
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