User blog comment:Flitri/An ordinal Collapsing up to the Least weakly Mahlo Cardinal/@comment-35470197-20190409053305/@comment-35470197-20190409145927

Thank you for the reply.

> 1) I meant to define F as a set of functions.

But you did not define so. What you wrote is that F is the set of the images of functions. Please read back your defintion.

> 2) that alpha parameter refers to an outside ordinal as in the case of the Omega indexed numbers.

I could not understand the precise meaning. Although Enum usually denotes an enumeration function of the given class, you just regard it as the image itself, right?

For example, what are the definitions of Enum{ x | x = Ω(α)} and Enum[min]{ x | x = Ω(α)}? (Here, P(x,α) is the formula "x = Ω(α)" with free occurrence of x and α.)

> 3) Distance in this case is not a metric just a informal statement on the relative size of each enumerated ordinal in respect to each other

For example, what is the precise definition of the distance between Ω(1) and Ω(2) in the class \(\{x \in \textrm{Ord} \mid \exists \alpha \in \textrm{Ord}, x = \aleph_{\alpha}\}\)?

> 4) L[v] is the associated limit ordinal with some other ordinal v, recall that N is the set of a Natural Numbers so little omega is the limit of that sequence.

But N is not a successor ordinal, but a limit ordinal, which trivially coincides with ω in ZF set theory. Even though you restrict v to a successor ordinal, why could you substitute N for v?

Also, please write down the precise definition of "the associated limit ordinal" with a given successor ordinal v.

> 7) That omega the the smallest regular fixed point of the x -> Omega(x) function, which the psi function never reaches akin to the first level psi function which never reaches Big Omega. It is indexed.

Then it is just the least weakly inaccessible cardinal. Since the convention Ω[1] is not the traditional one, it is better to clarify the meaning in the manuscript.

> 8) I am not quite sure why I can’t used that considering that Most ordinal collapsing functions use a similar means of getting to larger ordinals, could you explain that better?

OCFs (in mathematical paper) do no use such truth predicates. Recall that a statement itselt is not a term in set theory. Therefore in order to deal with a function admitting an input of a statement, we need to encode it by using formal languages with parameters in the way written in Keith J. Devlin, "Constructibility", Perspectives in Mathematical Logic, Volume 6, Springer-Verlag, 1984. It is well-known as a Tarski's theorem that there is no definable function admitting such encoded statements which values 1 if and only if the statement is true.

> 9) What exactly makes these rules ill-defined?

Well, I could not access the manuscript now, because the web page said "this document will be soon deleted". Therefore it is impossible for me to point out the exact rules... Do you know whether this problem will be solved soon or not?

Anyway, do you agree with the fact that expression-based definition is usually ill-defined, unless you set a canonical expression valid for all elements in the domain? If not, please read my introduction to OCFs.