What is the language of FOST (First Order Set Theory)

Question

I’ve been reading about Rayo’s number and I’m finding it difficult to grasp what exactly the language of FOST is. I understand the concept of finding the smallest finite number greater than any definable using $n$ symbols in a language, but I don’t know how the language Rayo used works. Anyone who could explain this would be very helpful, thanks.

Answer 1

The language consists solely of a single binary relation symbol, "$\in$" (together with the usual logical apparatus of first-order logic: parentheses, Booleans, quantifiers, variables, and equality).

• A side note: while we usually write "$\in$," when first learning set theory you should keep in mind that even though we're intending to talk about sets we're really talking about arbitrary structures satisfying some axioms, so it may make things clearer to write this symbol as the less-meaning-loaded "$E$."

You might object that this language is too small: where's $\cap$? Or $\cup$? Or $\mathcal{P}$? However, it's a good exercise to show that all the common set-theoretic operations can be described via elementhood alone, so we're really not losing anything here. (As to why we've chosen such a restrictive language, it makes a lot of arguments easier: induction on formula complexity is easier when there aren't many formulas!)

---

A more substantive objection kicks in when you realize that in set theory, everything is a set. This is related to the language issue, since it points to a certain "austerity" on the part of set theory that may initially be hard to motivate.

This sets-positive attitude seems to go wildly against normal mathematics (is $\pi$ a set?), and its appropriateness is indeed the subject of a lot of philosophical debate. However, that set theory is capable of "simulating" all of mathematics - regardless of how natural that simulation is - is pretty much beyond reproach; this is something you'll come to understand as you learn more about set theory (and in particular, a good starting point is to understand how arithmetic with natural numbers is faithfully captured by the finite ordinals).

That said, it would be very rude of me to not mention that there are perfectly adequate foundations for mathematics which do not adopt a sets-only ontology, and there are interesting philosophical and historical questions around why set theory developed as it did. But at this point we are getting very far afield.