User blog comment:P進大好きbot/New Googological Ruler/@comment-31580368-20190629142620/@comment-35470197-20190630074920

> Speaking of PTO, we can only talk about constructed models and structures below this, that can be proved by this theory.

It is just your own intuitive image irrelevant to the definition. I again ask the same question. Do you understand that there is a model of ZFC at which your functions have infinite loops under the assumption of consitency of ZFC?

> Stronger theories allow me to think about richer structures, such as Large countable ordinals, which I can use for example in OCF.

No. You do not have a valid recursive system of fundamental sequences. OCF is not a magical word. Even if you have a well-defined large ordinal, it does not generate a large number. I again ask the same question. Do you understand that a larger ordinal does not necessarily correspond to a larger function in HH even if you have a well-defined recursive system of fundamental sequences?

> I say that it is connected more with philosophy. If you do not believe in the existence of structures that you cannot build, then you are not a Platonist.

I did not mean "we do not know whether your functions can be constructed or not", but meant "the well-definedness directly contradicts mathematics". It is not related to Platonism. You could construct a proof of the ill-definedness if you actually understood what you wrote. That is why I wrote that you do not understand the definition of PTO.

> From this point of view, functions such as FOOT will actually be ill-defined.

FOOT is ill-defined from the beginning, because its well-definedness directly contradicts first order oodle theory...

I strongly recommend you not to use mathematical stuffs whose definitions you do not know. As you can see above, your explanations including existing examples contain so many mistakes because of the lack of the knowledge on their definition.