No subject
Soren Riis
sriis at fields.utoronto.ca
Sun Feb 1 11:30:13 EST 1998
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Categorical foundation axioms needs the touch of infinity
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First let me thank Colin McLarty for clarifying some
concrete questions to his axioms. Also I would like
to thank Vaughan Pratt (as well as others) for their
answer to a question I asked [Soren Riis, Tue 27 Jan 1998].
Not only did they answer my question, but they made me
want to learn more Category Theory.
I might be mistaken, but I think Colin McLarty's axioms
needs some addition. It might be a good idea to clarify
this before Friedman and other starts looking at the axioms.
All functions in McLarty's system are essentially got by
iterating basic operations like projections, embeddings,
quotient maps and pullbacks etc. - starting with N and a
few constant maps. These are the only kind of functions
McLarty's axioms seems to introduce. I think McLarty's
axioms needs some modification, otherwise we do not get
enough functions.
It also puzzles me that McLarty can do with only finitely
many axioms. Let me explain why I think this is a problem:
Any axiom-system which is strong enough to do analysis
must have a deductive strength at least that of ACA_0.
Now Peano's Arithmetic (the usual first order axiomatization
of N) do not have any extension which is finitely axiomatizable.
Thus ACA_0 (or any other system for analysis) must as far as
I can see contain infinitely many axioms.
Sometimes this fact is somewhat disguised because one is using
higher order logic and thus have smuggled in an infinite
axiom-schema in the underlying rules. McLarty did not specify
any rules for his system (so unlike the axioms for set-theory
[Harvey Friedman, Fri 23 Jan 1998] a computer wouldn't quite know
what to do with McLarty's axioms). I think we need to add some
sort of rule for set-formation to McLarty's axioms.
Any constructive suggestions?
McLarty's system still haven't got this little touch of
infinity!!
Soren Riis
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