# A strong plus-one hypothesis

Question. Is the following principle (or its weakening, with "for every real $$r$$" replaced with "for every real $$r$$ on a cone") consistent with ZFC?

$$(*)\quad$$ "For every normal functional $$F$$ of type $$n>2$$, there is some normal $$G$$ of type $$2$$ such that for every real $$r$$ we have $${}_1sc(F,r)={}_1sc(G,r)$$."

(Admittedly I don't know that the negation of $$(*)$$ is consistent, but $$(*)$$ seems so strong that an outright proof would seriously surprise me and so it seems better to ask for mere consistency.)

Above, "$$\le$$" denotes reducibility in the sense of Kleene's finite-type recursion (or its many equivalent formulations). For $$k\le n$$ and $$F$$ a type-$$n$$ functional, we let $$_ksc(F)=\{g\in tp(k): g\le F\}$$; this is the $$k$$-section of $$F$$. A type-$$n$$ functional $$F$$ is normal if it computes the type-$$n$$ "equality functional" $$^nE$$ (which determines whether a type-$$(n-1)$$ functional is identically zero or not); intuitively, $$^2E$$ is the usual Turing jump operator, and normality in type-$$2$$ is just "effective discontinuity".

The motivation for considering $$(*)$$ as even remotely plausible comes from (the $$k=1$$ case of) Sacks' Plus-One Theorem, which states that the result holds if we remove the real parameter. (Further motivation at least for the interestingness of $$(*)$$ comes from the extended plus-one hypothesis, but this does not seem to be directly related.)

My personal suspicion is that even the weak version of $$(*)$$ is inconsistent with ZFC, but I don't see how to prove that. Possibly a line of attack would come from examining the proof that the extended plus-one hypothesis fails in ZF+AD. However, as said above I don't see any direct connection here.

In the other direction, one approach to proving that $$(*)$$ is consistent would be to analyze a "boldface" version of the forcing Sacks used to prove the Plus-One Theorem. Roughly speaking, given a normal type-$$n$$ functional $$F$$ ($$n>2$$) Sacks forces with countable partial maps $$p:\omega^\omega\rightarrow\omega$$ satisfying two properties: $$(i)$$ that a certain hierarchy $$Trees(p)$$ of objects recursive in $$p$$ has a certain technical property, and $$(ii)$$ $$p\in{}_1sc(F)$$. (The generic object $$H$$ is then a functional defined on $$_1sc(F)$$, and the $$G$$ we want $$H\cup\{(f,0): f\not\in{}_1sc(F)\}$$.) The boldface version of this would probably consist of those $$p$$ such that for every real $$r$$ (possibly on a cone):

• $$(i)$$ The hierarchy $$Trees(p\upharpoonright {}_1sc(F,r), r)$$ has the technical property.

• $$(ii)$$ The restriction $$p\upharpoonright{}_1sc(F,r)$$ is an element of $$_1sc(F,r)$$.

However, it's not at all clear to me that this modified forcing behaves the way we would want.

• Incidentally, if we replace "$\le$" with "is r.e. in," we get the $k$-envelope; an analogous question can be formulated for envelopes with $1$ replaced by $2$ - see page $36$ of Longley's survey - and I would be equally interested in that, but for now I'm sticking with sections if only for simplicity. Dec 9, 2018 at 18:49
• Noah, did you ever put together a set of notes from your Generalized Recursion Theory course you were planning on teaching last year? I'd be interested in a copy if there is something available. Dec 13, 2018 at 16:04