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.

Mathematics is set theoreticaly based and set theory is an axiomatic
theory ,with undefined term -the "belongs to" relation letter in
addition to the logical connectives,equality and some discussion of the
rules of reasoning and I think rules for defining new terms and
predicates.As a start you could even look at Suppes -Introduction to
Logic (note the word mathematical is missing from the title) followed
by Suppes -Axiomatic Set Theory ,you would have a good start to an
adaquate axiomatic basis for Mathematics.Let me mention also the book
by J.Donald Monk -Introduction to set theory .He has a ten page
appendix called Axiomatic logic-"This appendix is devoted to a
rigourous development of a portion of mathematical logic sufficient to
found axiomatic set theory on....For a more thorough treatment of logic
using minimum mathematical apparatus we suggest Church - intro to
mathematical logic 1956."
Your looking in the wrong books.Most of the mathematical logic books
written in the last half of the 20th century set up there language
systems and did there analysis of them within set theory and even use
quite advanced mathematics to study the systems including the systems
used in ordinary mathematics studying
,consistency,independence,questions about limitations on decidability
etc.They were not interested in setting up the foundations of
mathematics.That was the main concern of a previous generation-
Dedekind,Frege,Russell,Zermelo etc.
Bourbaki -Set theory has just been translated to English but I havent
seen it.Perhaps they carry out the task adequately. Regards,Stuart M
Newberger

.

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