User blog comment:Boboris02/Breakthrough! Traditional OCF definition for TON/@comment-30754445-20180414193201

Was there anybody who claimed that TON isn't well-founded? I wasn't aware there was any controversy here. After all, it was already proven in "ZFC+measurable cardinal".

The only possible pitfall is that ZFC+measurable cardinal might turn out to be inconsistent. But the same can be said about the collapsing function approach. There are basically two possibilities here:

(1) Your collapsing function relies on "ZFC+measurable cardinal" or another similar axiom for it to work.

(2) Your collapsing function can be proven to work in a weaker system (say ZFC alone, or "ZFC+inaccessible cradinal").

If #1 is true then your collapsing function didn't solve anything. It cannot be any more robust than TON itself.

If #2 is true, then things get interesting. Assuming you've done everything correctly, then you've just improved the upper bound of TON's strength! After all, your collapsing function is meant to prove that TON is well-founded. So if your function can be shown to work without going beyond ZFC, then TON's Limit < PTO(ZFC).

Now, I'll be honest with you: I have absolutely no idea what you tried to do, and whether it actually works as such a proof. But if it does, then it is a logical necessity that one of the above two options is true.

And if #2 is true, I'd really like to know about it, because that would be a very important discovery indeed.