User blog comment:Bubby3/Notation provably equivalent to BMS pair sequences/@comment-35470197-20190530005229/@comment-35470197-20190601001122

> What if I prove the termination of the equlivant notation,

Then ok.

> or prove it equlivant to another notation?

It is ok only when you prove the termination of the notation.

> What if that notation people really haven't doubted the termination or well-foundness of, like R function up to {0,{0}}.

It is ok only when you prove the termination of the notation. Whether people doubt or not is not relevant to the provability.

> Will people start to doubt whether R function or other similar notations terminate, but they never did?

Oh... Are you seriously considering that if a notation is believed to be terminate, then it means that the temination is provable?

For example, remember the transcendental integer system, which assigns to each \(n\) the maximum of the outputs of Turing machines whose termination admits a proof of length \(\leq n\). It actually terminates for each standard input \(n\). But the termination at every \(n\) is not provable, although no one doubts the termination.

Provability is guaranteed only when you actually prove it or prove the provability of it. It heavily depends on what axiom you use, but not on whether people doubt or not under an intuition irrelevant to axioms.