User blog comment:Alemagno12/Ordinals in programming languages part 1: Brainfuck/@comment-24920136-20170404170242/@comment-30754445-20170404205927

I don't think any amount of FGH-like recursion would result in anything more powerful than a naive extension of Busy Beavers.

After all, whatever ordinal system that will be developed here, could always be emulated by a smallish ordinary brainfuck program (or an ordinary turing machine). And it seems, intuitively, that hardwiring the ordinal system into the language will - at most - give rise to a function about as strong as BB(Xn+Y) where X and Y are constants.

Creating a progamming language with ordinals would be cool for many other reasons, but I don't see how it would give rise to a strong uncomputable functions.

By the way, I wonder what would happen if we allowed running time to be an ordinal. If we could somehow define what a program's output after ω steps is, could this give rise to an uncomputable function which is substantially stronger than ordinary Busy Beaver?