User blog comment:Plain'N'Simple/A question for proof-theory experts/@comment-35392788-20191029194318/@comment-35470197-20191030124843

@Plain'N'Simple

> What if we added the condition that the fundamental sequences are strictly increasing?

Although you have answered it by yourself, what does "strictly increasing" mean here? If you require the property "α[n] < β[n] for any α < β", then usually the property does not hold.

@Syst3ms

> Anyway, a "reasonable FS system" should obey a few properties :

(As Plain'N'Simple pointed out, the first inequality seems to include a typo.)

> Most importantly, transfinite induction along \(\alpha\) is provable in some theory T iff \(\alpha[n]\) is provably total in T as well

The fourth condition (especially "if" part) rarely holds. For example, many of known fundamental sequences are primitive recursive, and their totality can be provable in many theories even if the domain of fundamental sequences are beyond their PTOs.