Is Set Theory Based on Circular Reasoning?

[gop]

Hi

I'm reading some stuff about proof theory and set theory right now and one question comes to my mind.

Set theory is defined in terms of FOL (First Order Logic). Nevertheless, when we "define" first order logic we already have the notion of a "domain of discourse", which is basically the same as a set. We also can't say "everything" is the domain of discourse because then we would need a universal set in set theory which doesn't exist (at least not in ZFC)
But then, we are defining one thing in terms of the other without knowing what the other is.

Isn't that sort of circular reasoning?

 

[MathematicalPhysicist]

Yes it is.

 

[gop]

well then the question would be isn't that a problem?

I mean how can we be sure that any proof is valid if we have to look at any quantifier and say okay that means "all x in the domain of discourse" but then we look up what it means that "x is in the domain of discourse" and we get another quantifier...

 

[CRGreathouse]

Sets are one of the basic undefined terms of mathematics. Anything fitting the characteristics of a set (contains objects not counting duplicates) can be considered a set. The same goes for point, line, etc.

 

[gop]

Yes that would sound logical to just say okay that is a set the same way as we say okay this is a predicate and it has to be either true or false (in FOL).

Hmm, so we have the primitive notion of a set to model the domain of discourse but we don't really say how that set can be constructed (Thus, Russels paradox i.e. is not a problem because we assume that we already have a well defined set).

Then, when we have FOL we build ZFC (which enables a rigerous treatment of how to construct well-formed sets) which then in turn enables us to constructs sets like the natural numbers etc..

Is this correct?