User blog comment:Emlightened/BIG FOOT is SMALLER than FISH NUMBER 7/@comment-1605058-20161225112922

First let me address the asterisked remark: yes, in FOOT we solely work inside the universe of sets, and all the "oodle" stuff indeed is just a fancy talk, the use of which I regret nowadays. I admit to all of that in [googology.wikia.com/wiki/User_blog:LittlePeng9/First_order_oodle_theory_-_clarification my recent blog post].

As for the claims within your blog post, I admit that the language is pretty much equivalent to the one in Feferman set theory, though this took me a short while to realize - for a moment I thought \(\text{Ord}_2\) is smaller than the second ordinal in Feferman's class \(C\), but then I have realized that we can use \(\text{Ord}\) as a parameter. (by the way, I think there might be some offset between \(\text{Ord}_\alpha\) and \(\alpha\)th element in \(C\), because when defining \(\text{Ord}_\alpha\) we add into the language symbols for all \(\text{Ord}_\beta,\beta<\alpha\), while in Feferman set theory we can't express all elements of \(C\) because there are too many of them, but I don't think it's a significant difference).

But even though I admit that, you are still wrong about FOOT being smaller than Fish number 7, since FOOT can also define FOST function, just as Feferman set theory can. Indeed, the truth predicate for V, at least for parameter-free formulas (or even ones with parameters from \(V_\alpha\)), is the same as the truth predicate for \(V_\alpha\) where \(\alpha=\text{Ord\}\)/\(\alpha\) is the first ordinal in the class \(C\), however you want to call it. This is because \(V_\alpha\) is elementarily equivalent to \(V\), so (by definition) the formulas satisfied by both are exactly the same, so we can define FOST function by refering to truth in \(V_\alpha\). Therefore we can define the "Rayo oracles" from Fish number 7, and indeed we can define that number in FOOT/Feferman set theory.

btw, you're a witch