FOM: Second order logic

[John Mayberry]

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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