User blog comment:Mush9/KING function/@comment-1605058-20170115211203

You are misusing the notion of a free variable. You shouldn't be saying that two free variables are equal, because they per se have no value. You should be speaking of the values assigned to the variables being equal.

I am not sure what you mean with the first sentence in the KING section. Is \(\alpha\) itself supposed to be a free variable, or \(x_\alpha\)? Also, unless you are going to refer to it itself later, why the need to introduce \(x\) as a single object? If you only want to talk about the variables \(x_\alpha\), it is pointless to invoke \(x\).

"we find the largest natural number ranks for all free variables of a particular KING statement." Again, free variables don't have a value, let alone a rank, so again you have to speak of value assignments. But then, are you only looking at a specific assignment, or at all possible assignments? (the latter sentence suggests the latter, but it's not explicitly stated)

Lastly: indeed, there is no need for the existential quantifier - you can speak of the ranks arising without adding \(\exists\) to the language. But the language without \(\exists\) is extremely weak. There is barely any meaningful set-theoretical statement you can do without quantifiers. For example (with me writing \(x_0\) for some fixed free variable), how do you write the statement "\(x_0\) is empty"? Or "\(x_0\) is an ordinal"? Or "\(x_0\) is the tenth busy beaver number"? You can't do any of that without a quantifier in your language.