Re: question about axiomatic set theory

Dani wrote: 
I already know the axiomatic development of set theory. So my question
might seem absurd - I don't see how that development is possible. My
question - how IS it possible?

Where this absurdity comes from: when I try to re-gain my
understanding, logic and set-theoretic approaches tangle together, and
trying to put one in front of the other just makes the problem worse. I
don't see how set theory can be axiomatically developed, if structures
are sets, and so are formal systems. As though it were not enough that
my thinking tangles set and logic, I began recently to read Nonstandard
Analysis by Abraham Robinson. Here, constructions like enlargements and
ultraproducts just invite the problem to get worse.

I hope somebody might have some idea about this. But it seems to me
that maybe mathematics is just ambiguous to some extent - which can
mean that not everything in it can be stated on paper, but it open to
individual interpretation. But that would seem to be a contradiction,
given how universally accepted the mathematical theories are, despite
their complexity.

My feeling is that at some level you have to be a platonist, that is, you have to accept that there is some kind of reality to all this stuff, which axiom systems are not defining, but describing. At some level, you have to assume a kind of naive set theory (as opposed to an axiomatic set theory) to get off the ground. I learned this subject from the book by Mendelson (spelling?), and before he even starts with the axiomatic stuff he give an introduction to naive set theory (well at least in the edition which I read, which I suspect is fairly old).

In any case, most working mathematicians don't study the axiomatic foundations very much, and indeed, I think quite a few universities only teach it at a very light level (maybe in the first week or so of real analysis/measure theory, mostly covering concepts like countable sets).

My speculation is that sometime in the future, our current scheme will be found to have some dreadful contradiction.

Stephen
.

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