[FOM] Question about Set Theory as a formal basis for mathematics

[Andrea Proli]

Hello everyone,
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  
recursion.

So, the semantics of ZF is given in terms of what ZF itself defines? Or am  
I simply confused?

Thank you in advance,

Andrea