User blog comment:Eners49/A whole new superclass of infinities?/@comment-35470197-20180722021627/@comment-35470197-20180724094128

> 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.

Hmm. Recall that mathematicians do not necessarily clearly understand ordinals beyond \(2^{\aleph}\) even though almost all branches of mathematics are based on set theory. It does not mean that they do not know set theory.

Therefore I thought that it is not so strange if the OP has several misunderstandings on infinity even if I assume that the OP knows set theory at university-student-level.

Moreover, we often use jargons before fully understanding what we should care about. For example, mathematicians often use (non-small) categories and \(V\) without knowledge of NBG or inaccessible cardinals. At least, the number of people who know \(\varphi(4,0)\) is much less than that of people who know \(\textrm{ON}\), I think. Of course, it depends on community.

Anyway, I seem wrong as a result. At least in this community, the number of people who know \(\varphi(4,0)\) would be greater than that of people who know \(\textrm{ON}\). Since I am a newbie in this community, I failed in estimating the level of an appropriate answer. It is my failure.

> 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.

Uh...

In Japanese googology community, in which I am very very very\({}^{\omega}\) weak compared to professionals there, I heard that there were many professionals powerful in analysis far beyond Japanese community. So I would like to believe them. I never doubt other people without any appropriate counter-proofs. Since I do not use results on analysis which I have not verified by myself, it would not matter.

Anyway, thank you for the information. You are so honest.

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

As I said, I guessed that the "largest ordinal" does not mean a real ordinal. If I had thought that it meant a real ordinal, then I would have explained the problem on "add 1". Since I thought that it was something which is not an ordinal, I could not define "add 1".

Summary: Does it make sense?
 * 1) I guessed that the OP was considering the case \(\infty + 1 = \infty\).
 * 2) Then I guessed that the "largest ordinal" does not mean an ordinal.
 * 3) Then I could not use "+1" in the explanation, even though I started from the guess \(\infty + 1 = \infty\).
 * 4) At least, I could say that there is no object containing all ordinals.

Also, the reason why I used "unprovable" but not "disprovable" is just because I thought that the former word is well-known to people who are not logicians. I should have used "disprovable" in order to make the statement clearer.