[FOM] Question about Set Theory as a formal basis for mathematics
aprol at tin.it
Sat Feb 25 09:23:45 EST 2006
I am a newbie here, I have not a deep knowledge of mathematics because it
is not the primary subject of my studies. However, my personal interests
brought me to an effort in understanding the very foundations of
mathematics, which I assume to be (most say) Set Theory.
There is a question I would like to ask this mailing list about ZF Set
Theory, and all other Set Theories in general. The question is: are they
really stable, formal foundations for mathematics?
I mean: as far as I know, ZF is a first-order theory, and first-order
theories have a standard denotational, model-theoretic semantics. In model
theory, symbols are given an interpretation in terms of sets and relations
(which are also sets). Isn't this a circular definition?
The semantics of sets is defined in terms of sets, and this recursive
definition does not seem to be explicited (kind of a "fixpoint" definition
would be more comprehensible to me...)
This is quite different from a mere axiomatization: I can accept that sets
are not defined in terms of anything else because they are the
foundational element of mathematics, but it seems somehow "wrong" to me
that they are defined in terms of themselves, in such an implicit
So, the semantics of ZF is given in terms of what ZF itself defines? Or am
I simply confused?
Thank you in advance,
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