User blog:TheKing44/The Grand Finale Functions

Rayo's function grows faster than any function definable in first-order set theory. However, it would be incorrect that this function settles Googology in first-order set theory, since it is not itself definable in that language. It would only be fair to compare it to other functions in second-order set theory.

In this post, I will define a family of extremely fast growing functions that can be defined in first-order set theory. First, some prerequisites.

A satisfaction predicate tells you which arguments satisfy a formula. More formally, for a satisfaction predicate $$\#$$, we must have $$\forall x,y,\dots,z. \#('\phi', (x,y,\dots,z)) \iff \phi(x,y,\dots,z)$$ for every formula $$\phi$$.

We can not define a satisfaction predicate in first order set theory that works for every formula in first order set theory. We can however define a satisfaction predicate over a subset of such formulas.

We can define a $$\#$$ for formulas that only contain bounded quantifiers. Given a formula $$\phi$$ with arguments $$x,y,\dots,z$$, we first pick some set that contains $$x,y,\dots,z$$ and is transitive. This set will function as our universe. We will call it $$U$$.

Now, we define a function that takes a formula $$\phi$$ (only containing bounded quantifiers), tuples over $$U$$