User blog comment:Eners49/A whole new superclass of infinities?/@comment-35470197-20180722021627/@comment-30754445-20180724082504

"Uh. Ok. I am very sorry for that. I really thought that the OP was using set theory (with several misunderstandings on infinity)"

True, he was using set theory without realizing it. Most of the people here do that.

But the "with several misunderstandings on infinity" part should have told you that he doesn't really understand set theory. The OP was clearly unaware of the fact that there are uncountable infinities within set theory, for example. Given that, it should have been obvious that an answer using technical jargon will not help the OP at all.

"Thank you for pointing out that very few googologists are familiar with set theory. I did not know the fact at all, because many googologists use the PTO of KP set theory and also I was suggested to use NBG set theory instead of ZFC set theory in my blog post. Since even mathematicians do not necessarily know KP set theory or NBG set theory unless they are logicians or categorists, I thought that many googologists here are unbelieavably familiar with set theories."

It gets especially crazy when people start discussing weakly compact cardinals and reflection principles, or claim that certain notations reach the PTO of Z2 :-)

But I've got news for you: Most of this is just a facade. The vast majority of the people who use these terms here, have absolutely no idea what they're doing. They just fake it and hope nobody (least of all themselves) will notice.

BTW we used to have a few very knowledgable pros here, but most of them left in the past few months. It's a shame, really.

"Yeah. You are completely right. I would like to add why I wrote this. I though that the OP was considering the property like \(\infty + 1 = \infty\) as in the real analysis (without contradiction). Then I guessed that the "greatest ordinal" does not actually mean an ordinal, but means an object "beyond" all ordinals in this context."

Yup. That's what the OP meant.

Which still doesn't explain why you wrote "the existence of the largest ordinal number is unprovable under ZFC". :-)

As for explaining the Burali-Forti paradox, the phrase "you can always add 1" does that quite nicely. Anybody who has even the tiniest experience of using ordinals in googology, can easily understand why "adding 1" always gets you a larger ordinal. It's one of those rare cases where set theory actually lines up with our everyday intuition.