I suspect it’s possible to come up with a definition of an oracle which corresponds to Mahlo. Of course, to actually work out the details I’d have to understand Mahlo itself, which I suspect is a ‘super-generalization’ of ‘you see that sequence? generalize it’.

If I ever find myself with nothing to do for a few months I’d really like to become well versed in this stuff, unfortunately the chances of that happening are pretty slim. Until then, I’m just a dilettante.

Come to think of it, any oracle corresponding to mahlo would have to have a way of describing the machines it’s plugging into the oracle, so if mahlo is inconsistent it would just result in a potential circularity in the repeated subtraction, so the question is still mathematically well-defined, it just might not halt because of infinite recursion.

]]>(1) “And so on, creating an infinite hierarchy.”

(2) “See that infinite hierarchy? Define a new machine that can query it anywhere.”

To get to uncountable ordinals, we’d have to stipulate a *new* operation that went beyond (1) and (2).

Once you throw that away (say, by going to uncountable ordinals), the sentences you construct *might* still have definite truth-values, but I’ll no longer bet my life on it. ðŸ™‚