Strongly relativizing reals

Abstract: For a real $f\in \mathbb{N}<sup>\mathbb{N}$, it is in general not the case that every set which is both $Σ<sup>1_1(f)$ and $Π<sup>1_1(f)$ is the $f$-section of a $Δ<sup>1_1$ set. However, there are reals $f$ for which this, in fact, does happen; we say that such a real strongly relativizes $Δ<sup>1_1$. In this paper, we will prove that the reals which strongly relativize $Δ<sup>1_1$ are exactly the hyperlow reals, i.e., the $f\in \mathbb{N}<sup>\mathbb{N}$ with $ω<em>1<sup>f=ω_1<sup>{\text{ck}}$. We will also study reals that strongly relativize the classes $Δ⁰</em>α$ for $1\leq α&lt;ω<em>1<sup>{\text{ck}}$. We characterize the reals that strongly relativize $Δ<sup>0_1$ subsets of $\mathbb{N}$ as the reals which are computably dominated. For $1\leq α&lt;ω_1<sup>{\text{ck}}$, we show that there are continuum many reals which strongly relativize $Δ⁰</em>α$ subsets of $\mathbb{N}$. We will also find non-trivial examples of reals which strongly relativize $Δ<sup>0_α$ ($1\leq α&lt;ω_1<sup>{\text{ck}}$) subsets of Baire space.

• The paper characterizes exactly the reals that strongly relativize Δ¹₁ as the hyperlow reals, those satisfying ω₁ᶠ = ω₁ᶜᴷ, and connects this property to uniform Δ¹₁ coding.
• For Δ⁰₁, strong relativization at Baire space is equivalent to computable domination and truth-table irreducibility, while examples involving Π⁰₁ singletons show that the property need not be preserved under Turing equivalence.
• At higher arithmetical levels, Δ⁰₁₊ₐ-dominated reals and Π⁰₁₊ₐ singletons provide continuum-many sufficient examples, but domination is not necessary and several exact characterizations remain open.

The paper studies which parameters $f \in \mathcal{N} := \mathbb{N}^\mathbb{N}$ have the property that every set in the relativized self-dual pointclass $\Delta(f)$ is in fact the $f$-section of a lightface $\Delta$ set. This property, called strongly relativizing, captures when the naive relativization of a self-dual Kleene pointclass — defined as $\Gamma(f) \cap \neg\Gamma(f)$ rather than as sections of $\Delta$ sets — coincides with the "honest" section pointclass $\mathcal{S}\Delta(\mathcal{N}; f)$. The paper gives complete characterizations for $\Delta^1_1$, partial characterizations for $\Delta^0_1$ and $\Delta^0_{1+\alpha}$, and establishes several structural consequences concerning Turing equivalence and coding of definable sets.

Background and setup

Relativization is the standard bridge between effective (lightface) and classical (boldface) descriptive set theory: for a pointclass $\Delta(f)$0 on recursively presented Polish spaces, $\Delta(f)$1 consists of $\Delta(f)$2-sections $\Delta(f)$3 of $\Delta(f)$4 sets $\Delta(f)$5. For non-self-dual classes such as $\Delta(f)$6 or $\Delta(f)$7, this process recovers exactly the boldface class. For self-dual classes $\Delta(f)$8, however, one defines $\Delta(f)$9, and while every $f$0-section of a $f$1 set lies in $f$2, the converse can fail.

A real $f$3 strongly relativizes $f$4 at a space $f$5 if

$f$6

and strongly relativizes $f$7 if this holds at every recursively presented Polish space. Two preliminary observations streamline the theory: for every self-dual Kleene pointclass other than $f$8, using Cantor space or Baire space as the parameter space yields the same section classes; and for $f$9 restricted to effectively zero-dimensional spaces (which includes all spaces considered), compactness of $\Delta$0 again makes the two coincide. A further reduction lemma shows that if $\Delta$1 strongly relativizes $\Delta$2 at $\Delta$3 and there is a computable surjection $\Delta$4 with computable right inverse, then $\Delta$5 strongly relativizes $\Delta$6 at $\Delta$7 — so strong relativization is invariant under computable isomorphism and transfers from $\Delta$8 to $\Delta$9.

The analytical case: hyperlowness

The central result identifies the reals strongly relativizing $\Gamma(f) \cap \neg\Gamma(f)$0: they are exactly the hyperlow reals, i.e., those $\Gamma(f) \cap \neg\Gamma(f)$1 with $\Gamma(f) \cap \neg\Gamma(f)$2. The forward direction uses the reduction property for $\Gamma(f) \cap \neg\Gamma(f)$3 together with the hyperlow characterization that every $\Gamma(f) \cap \neg\Gamma(f)$4 set containing $\Gamma(f) \cap \neg\Gamma(f)$5 has a $\Gamma(f) \cap \neg\Gamma(f)$6 subset still containing $\Gamma(f) \cap \neg\Gamma(f)$7: given $\Gamma(f) \cap \neg\Gamma(f)$8, one reduces two complementary $\Gamma(f) \cap \neg\Gamma(f)$9 definitions to obtain a $\Delta$0 set $\Delta$1 containing $\Delta$2 whose members code complements, shrinks it to a $\Delta$3 set $\Delta$4, and glues the pieces into a single $\Delta$5 relation whose $\Delta$6-section is $\Delta$7. The converse is a diagonalization: if $\Delta$8 is not hyperlow then $\Delta$9, so $\mathcal{S}\Delta(\mathcal{N}; f)$0 computes an enumeration $\mathcal{S}\Delta(\mathcal{N}; f)$1 of a good $\mathcal{S}\Delta(\mathcal{N}; f)$2 coding $\mathcal{S}\Delta(\mathcal{N}; f)$3 of the $\mathcal{S}\Delta(\mathcal{N}; f)$4 subsets of $\mathcal{S}\Delta(\mathcal{N}; f)$5; the set $\mathcal{S}\Delta(\mathcal{N}; f)$6 is then $\mathcal{S}\Delta(\mathcal{N}; f)$7 but provably not an $\mathcal{S}\Delta(\mathcal{N}; f)$8-section of any $\mathcal{S}\Delta(\mathcal{N}; f)$9 set, since any purported index $\Delta^1_1$0 contradicts itself at positions where $\Delta^1_1$1.

Two corollaries follow immediately. First, since there are continuum-many hyperlow reals, continuum-many reals strongly relativize $\Delta^1_1$2 at every space — notably including non-effectively-zero-dimensional ones. Second, hyperlowness is equivalent to the statement that whenever $\Delta^1_1$3, there is a total $\Delta^1_1$4 function $\Delta^1_1$5 with $\Delta^1_1$6. This is explicitly analogous to truth-table irreducibility, an analogy made precise in the next section.

For higher analytical classes $\Delta^1_1$7, $\Delta^1_1$8, the paper obtains only conditional results. Under Projective Determinacy, $\Delta^1_1$9 has the reduction property, so the same proof shows that any real satisfying the appropriate $\Delta^0_1$0-separation analogue of hyperlowness strongly relativizes $\Delta^0_1$1, while any real strongly relativizing $\Delta^0_1$2 cannot compute the corresponding $\Delta^0_1$3-complete set of integers. Whether these conditions are equivalent, whether uncountably many such reals exist, and whether the answers depend on determinacy axioms are left open.

The recursive case: computable domination

For $\Delta^0_1$4 at $\Delta^0_1$5, strong relativization coincides with classical notions from computability theory. The key lemma is that $\Delta^0_1$6 is an $\Delta^0_1$7 subset of $\Delta^0_1$8 if and only if $\Delta^0_1$9; the nontrivial direction uses compactness of $\Delta^0_{1+\alpha}$0 to extract a computable modulus from a computable clopen definition. Consequently:

$\Delta^0_{1+\alpha}$1

where the last equivalence is due to Soare's exposition of classical results. By Martin–Miller, there are continuum-many computably dominated reals in $\Delta^0_{1+\alpha}$2 (relativizable to any base), so continuum-many reals strongly relativize $\Delta^0_{1+\alpha}$3 at $\Delta^0_{1+\alpha}$4; moreover no noncomputable real Turing comparable with $\Delta^0_{1+\alpha}$5 is computably dominated.

The picture changes for Baire space. Every computably dominated $\Delta^0_{1+\alpha}$6 does strongly relativize $\Delta^0_{1+\alpha}$7 at $\Delta^0_{1+\alpha}$8 (via its canonical graph encoding into Cantor space), but the converse fails: every $\Delta^0_{1+\alpha}$9-singleton $\Delta(f)$00 strongly relativizes $\Delta(f)$01 at $\Delta(f)$02, by a tree-based argument that converts a $\Delta(f)$03 definition into a $\Delta(f)$04 relation using the unique-path property of the computable tree isolating $\Delta(f)$05. Since some $\Delta(f)$06-singletons are Turing equivalent to $\Delta(f)$07, and $\Delta(f)$08 is not computably dominated, the paper concludes that strongly relativizing $\Delta(f)$09 at $\Delta(f)$10 is not preserved under Turing equivalence — a sharp contrast with all other self-dual Kleene pointclasses considered, for which Turing equivalence does preserve the property (the case $\Delta(f)$11 remains open). The same example yields a real strongly relativizing $\Delta(f)$12 at $\Delta(f)$13 but not at $\Delta(f)$14.

An attempted generalization via a higher-type notion — computably Baire-dominated, requiring every $\Delta(f)$15-computable total functional $\Delta(f)$16 to be pointwise dominated by a computable one — collapses: an appendix presents Patrick Lutz's proof that the only computably Baire-dominated reals are the computable ones. The proof combines a tree-extension characterization of Baire domination with a partition of $\Delta(f)$17 into two $\Delta(f)$18 sets neither of which contains an infinite $\Delta(f)$19 set, forcing any such $\Delta(f)$20 into a finite $\Delta(f)$21 set and hence making it computable. Whether non-$\Delta(f)$22 reals strongly relativize $\Delta(f)$23 at $\Delta(f)$24, and whether there are continuum-many, remain open.

The arithmetical hierarchy

For $\Delta(f)$25 with computable ordinal $\Delta(f)$26, the paper exploits the standard identity $\Delta(f)$27, where $\Delta(f)$28 is the iterated Turing jump. However, the analogous identity fails for the section classes: if $\Delta(f)$29 strongly relativizes $\Delta(f)$30 at $\Delta(f)$31, then $\Delta(f)$32 is a proper subclass of $\Delta(f)$33. This failure reflects the fact that $\Delta(f)$34 definitions over Baire space involve the jump $\Delta(f)$35 jointly in parameter and variable, preventing a clean transfer of the $\Delta(f)$36 results.

Two sufficient conditions are established. First, $\Delta(f)$37 is $\Delta(f)$38-dominated if every total $\Delta(f)$39 function $\Delta(f)$40 is pointwise bounded by a total $\Delta(f)$41 function; equivalently, $\Delta(f)$42 is computably dominated relative to $\Delta(f)$43. Any such $\Delta(f)$44 strongly relativizes $\Delta(f)$45 at $\Delta(f)$46, via a bounded-search argument using the closure of $\Delta(f)$47 under bounded quantification against $\Delta(f)$48 functions. Combining Martin–Miller with MacIntyre's extension of the Friedberg jump inversion theorem yields continuum-many $\Delta(f)$49-dominated reals, hence continuum-many reals strongly relativizing $\Delta(f)$50 at $\Delta(f)$51.

Second, every $\Delta(f)$52-singleton strongly relativizes $\Delta(f)$53 at every space, generalizing the $\Delta(f)$54 singleton argument through the normal form for $\Delta(f)$55 sets on arbitrary recursively presented Polish spaces. This second condition is strictly broader than domination: taking $\Delta(f)$56 to be a $\Delta(f)$57-singleton Turing equivalent to $\Delta(f)$58 produces a real that strongly relativizes $\Delta(f)$59 everywhere yet is not $\Delta(f)$60-dominated, witnessed by the convergence-time function for $\Delta(f)$61-computations, which is $\Delta(f)$62-computable but unbounded by any $\Delta(f)$63-computable function. Thus domination is sufficient but not necessary, and no natural necessary-and-sufficient condition is known.

On the negative side, any $\Delta(f)$64 computing Kleene's $\Delta(f)$65 fails to strongly relativize any $\Delta(f)$66 at $\Delta(f)$67, by the same diagonalization against a $\Delta(f)$68 parameterization used in the hyperlow argument. As a final application, the paper proves there is no nice coding of the $\Delta(f)$69 subsets of $\Delta(f)$70 — no $\Delta(f)$71 index set with uniformly $\Delta(f)$72 dual representations — since such a coding would let a suitably chosen $\Delta(f)$73-singleton enumerate indices and reproduce the diagonal contradiction.

Limitations and open questions

Several gaps remain explicit in the paper. The characterization of strongly relativizing $\Delta(f)$74 relies only on the reduction property and good codings, so it extends under PD to odd levels of the analytical hierarchy, but no characterization is given for even levels, and the existence of uncountably many reals strongly relativizing $\Delta(f)$75 for $\Delta(f)$76 is open, possibly independent of ZFC. For the arithmetical side, the exact strength of $\Delta(f)$77 strong relativization on Baire space is unresolved: it is unknown whether non-$\Delta(f)$78 reals can have the property, whether continuum-many do, and whether any real strongly relativizes $\Delta(f)$79 at $\Delta(f)$80 without doing so at $\Delta(f)$81. The status of $\Delta(f)$82 under Turing equivalence is also undetermined, and no intrinsic necessary-and-sufficient condition replaces $\Delta(f)$83-domination, which is shown to be non-necessary.

Conclusion

This paper clarifies when the relativization of a self-dual Kleene pointclass is faithful to its sections. The answer is complete for $\Delta(f)$84 — precisely the hyperlow reals — and structurally illuminating for $\Delta(f)$85, where strong relativization at $\Delta(f)$86 coincides with computable domination and $\Delta(f)$87-irreducibility, yet behaves anomalously on Baire space by failing to be Turing invariant. For higher arithmetical levels, continuum-many examples exist via both domination and singleton constructions, though neither condition is necessary, and the absence of nice codings of $\Delta(f)$88 sets underscores a genuine obstruction to uniform effective parametrization of these classes.