User blog comment:JHeroJr/Can Someone Tell Me How To Create An Uncomputable Function That Is Not Naive?/@comment-35470197-20190623220959

> And if a function were defined "f(n) = the smallest number not computable to a level-N turing machine", wouldn't all the values be undefined?

What is N? Is it n? Also, what is "level-N"? Is it "N-th order oracle"? In order to create an uncomputable large number, you need to fix an axiom of set theory, and precisely describe the definition in a formalised way in the theory.

I strongly recommend you not to use natural-language-based explanations, because the formalisation is not unique. Whether it is well-defined or not depends on the formalisation.

First of all, what axiom do you assume? This is a starting point. For example, if you want to work in ZFC set theory, it is not reasonable to compare your numbers with Rayo's number.