User blog comment:DontDrinkH20/H-Boogol-Boogol: Hopefully a very big non-salad number/@comment-35470197-20180819021045/@comment-35470197-20180821071728

> first question

Oh, I see. You are refining the formal language itself. Smart.

> second question

I see.

> If I made it all definable classes though, would it work with normal specification?

I do not know. Could you tell me why you needed \(\mathfrak_a(b)\)? Isn't it bounded by \(\mathcal{H}_a-\textrm{Pre}_{\mathcal{N}}(b)\), because \(\mathcal{H}_a(b)\) is described as \(S^{\mathcal{H}_a-\textrm{Pre}_{\mathcal{N}}(b)}0\) in any model of ZF? Then you can simply use \(\mathcal{H}_{10 \uparrow^{100} 10}-\textrm{Pre}_{\mathcal{N}}(10 \uparrow^{100} 10)\) without using the satisfaction at all definable class-sized models.

Maybe I am wrong, because I could not imagine why you and Emlightened needed forcing here.

At least, if you use all (definable) class-sized models, then I guess that the resulting number is not larger than Rayo's number. Rayo did not refer to the satisfaction at ALL definalble class-sized model, but a SINGLE unspecific set-sized model. As I wrote above, using truth at a set-sized model is very strong, because the satisfaction at all definable class is purely proof-theoretic.

For example, consider the formula \(\varphi\) given as \(\aleph = \alpha_X\). The pair \((\textrm{ZF},\varphi)\)is not a defining pair in your sense. On the other hand, for a set-sized model \(M\) satisfying \(\varphi\), Rayo's number refering to \(M\) is greater than \(x\) corresponding to \(X\) named by \(\varphi\).

The point is that Rayo's number is not definable at \(M\), but is definable at \(V\) as additional unspecific data of \(M\), which is not definable at \(V\). (So Rayo's number itself is not well-defined if we are only allowed to use specific well-defined obeject, as my personal opinion. For more detail, see my blog post.)

On the other hand, in your definition with \(T = \textrm{ZF}\), you are only referring to definable natural numbers, because you are considering all models. Since you quantified models, you have no problem on the use of an "unspecific" model. So your definition is fair in a rule that you are allowed only specific well-defined objects, which Rayo's number does not obey.

> Thank you for these very informational comments!

You're welcome. I am happy to read such a cool approach. I like your number, even though I have not completely understood the definition yet :D

By the way, I am confused because the notion of the \((a,b)\)-representability is defined in a way independent of \((a,b)\). Maybe there is some typo. (Any additional assumptions such as \(|C| < a+1\) and \(|P| < b+1\)? How about \(F\)?)

Also, \(\mathfrak{H}_a-\textrm{Pre}_{\mathcal{N}}(b)\) might be a typo of \(\mathcal{H}_a-\textrm{Pre}_{\mathcal{N}}(b)\).