Kenneth Kunen in his “The Foundations of Mathematics” writes:
- ‘Set theory is the study of models of ZFC’ (p. 7)
- ‘Set theory is the theory of everything’ (p. 14)
With (1) Kunen is pointing to a change in the intended use of the axioms of ZFC: ‘there are two different uses of the word “axioms”: as “statements of faith” and as “definitional axioms”.’ (p. 6).
With (2) he means ‘set theory is all-important. That is
- All abstract mathematical concepts are set-theoretic.
- All concrete mathematical objects are specific sets.’ (p. 14)
According to (1), to be a set is to be any of the individuals of the universe of a particular model of ZFC, just like being a numeral (standard or not) is being any of the individuals of the universe of a particular model of PA (here I am using Shoenfield’s terminology in “Mathematical Logic”, p. 18).
But, according to (2), models are sets too, as any other objects dealt with in the metatheory.
What's more, models of set theory are defined in terms of relative interpretations of set theory into itself, a syntactical concept. (See Kunen’s “Set Theory. An Introduction to Independence Proofs”, p. 141), which makes the whole thing a bit more confusing.
The view of the axioms of set theory as "definitional axioms" is appealing. And more in regard of (2) since then they pretend to define all that there is. The study of models of set theory has an intrinsic interest, but why reduce the study of set theory to it? Or stated another way, why abandoning the old view?
I would like to know if set theorists do stick to one view or another or shift comfortably between both at need, and the reasons they have to do so.