The arithmetic hierarchy is naturally extended to all ordinals via ordinal notations creating a hierarchy for all hyperarithmetic sets. The polynomial time hierarchy is defined analogously to the arithmetic hierarchy but I've never seen it extended into the transfinite.

Given that every ordinal below $\omega_{CK}$ is the height of some polynomial time relation does it make sense to extend the polynomial time hierarchy to the transfinite or does something fundamentally break? If so what? Does it collapse after level $\omega$? Or is this a thing and I just haven't heard of it?

--

It seems like one could just define a polynomial time ordinal notation and use that to define a polynomial time analog to the sets $0^\alpha$ but I'm interested in any natural way of extending the analogy.

The arithmetic hierarchy is naturally extended to all ordinals via ordinal notations creating a hierarchy for all hyperarithmetic sets. The polynomial time hierarchy is defined analogously to the arithmetic hierarchy but I've never seen it extended into the transfinite.

Given that every ordinal below $\omega_1^{CK}$ is the height of some polynomial time relation does it make sense to extend the polynomial time hierarchy to the transfinite or does something fundamentally break? If so what? Does it collapse after level $\omega$? Or is this a thing and I just haven't heard of it?

--

It seems like one could just define a polynomial time ordinal notation and use that to define a polynomial time analog to the sets $0^\alpha$ but I'm interested in any natural way of extending the analogy.