User blog comment:Boboris02/Breakthrough! Traditional OCF definition for TON/@comment-30754445-20180414193201/@comment-30754445-20180415121842

If the equivalence is as intuitive as you claim it is, then it is very very unlikely to be "unprovable" in ZFC. Any intuitive argument you could make would probably be translatable into a formal proof in ZFC (and a far weaker theory would probably suffice as well).

After all, the whole point of ZF/ZFC was to formalize our intuition about sets into an axiomatic system. And in actuality, ZF/ZFC is already strong enough to tell us that our intuitive assumptions lead to quite counter-intuitive results (the axiom of choice, for example, is pretty much a "damn if you do, damn if you don't" proposition, when it comes to satisfying our intuition).

So trying to use "intuition" to go beyond standard set theory is not likely to be fruitful.