User blog comment:Boboris02/Ordinal Analysis of Theories/@comment-11227630-20180213035846/@comment-30754445-20180213212307

What?

How can a computer program provide evidence that there are no infinite loops, when the number of steps in most cases will be googologically large?

There's no way to distinguish between a program that terminates after 10^^^10 steps and a program that enters a loop after 10^^^10 steps. So computers are basically useless on this front (and this means, btw, that you've probably misunderstood the purpose of Taranovsky's program. Surely he knows what I've just said, so there's no way he is resorting to computer programs as evidence on this front).

At any rate, my question stands:

What evidence do we have for TON being well-founded ("well-founded" simply means that it is a well-defined ordinal notation with no gaps or loops)?

I assume such evidence does exist: Either there's an actual formal proof in some widely accepted axiomatic system (like ZF) or there's a solid intuitive explanation.

Either way, it is virtually impossible for such evidence to be independent from ZF. This is exactly why I'm asking this question: The very evidence that shows that TON is well-founded can be used to provide an upper bound to TON's strength.

So I ask again:

What evidence do we have for TON being well-founded?