We extend results found by Greenberg, Turetsky, and Westrick in [7] and investigate effective properties of bases of uncountable free abelian groups. Assuming V = L, we show that if κ is a regular uncountable cardinal and X is a ∆11(Lκ) subset of κ, then there is a κ-computable free abelian group whose bases cannot be effectively computed by X. Unlike in [7], we give a direct construction.
We develop a new metatheorem for 0-priority constructions for transfinite ordinals η. Our metatheorem simplifies previous metatheorems by Ash, Ash–Knight and the author. A drawback of this simplification is that its applications are more limited: It can only be used when the object being built is an ω-presentation of a structure. However, in practical terms, this is not a huge limitation and the majority of proofs in the literature that use the Ash–Knight metatheorem can be done using our metatheorem. Often in computability theory we need to build a computable object using non-computable information — sometimes even non-computable information about the very same object we are building. The main tool for such constructions is the priority method, which has become increasingly more involved and sophisticated since the 1950s. Priority arguments are classified in terms of how much non-computable information is needed to verify the construction. The most common priority constructions are the finite-injury ones [Fri57, Muc56]. They are used when the information needed is 0′-computable. Infinite-injury priority constructions [Sho61, Sac63] are used when 0′′-computable guesses are needed. There are various zero-triple priority constructions [Lac76] in the literature, but they are very complicated, and far less common. Beyond that point, humans cannot keep track of the combinatorics anymore. Well, that is unless the level-by-level combinatorics of the proof is uniform and one can describe the work done at all the levels by a single procedure. There have been various proposals for general 0(n)-injury constructions: Harrington’s worker’s method [Har76], Lempp and Lerman’s trees of strategies [LL95, Ler10], Ash’s [Ash86] and Ash–Knight’s [AK00] η-systems, and Montalbán’s systems of true stages [Mon14]. Among these methods, the Ash–Knight’s η-system is the only one that contains a clear metatheorem, where if a certain combinatorial machinery can be put in place, one can then apply the metatheorem as a black box and produce the desired computable object out of ∆η-information. The proof of the metatheorem is complicated. But the whole point of the metatheorem is that one does not need to know its proof to use it. A disadvantage for the metatheorem is that it is applicable in a limited number of situations. It has only been used in computable structure theory, where the combinatorial features needed for the metatheorem often occur naturally. Let me emphasize, though, that despite its limitations, it has been extremely useful in computable structure theory, and has been applied to a wide range of situations. In this paper we propose a new metatheorem. The advantage of the new metatheorem is that it is much easier to use than the Ash–Knight’s metatheorem. The disadvantage is that it is more limited. However, in most of the constructions from the computable-structuretheory literature where Ash–Knight’s metatheorem is useful, our metatheorem is too. So the restriction in applicability does not seem to be too limiting in practice, while the simplification Saved: December, 2019 Compiled: December 9, 2019 The author was partially supported by NSF grant DMS-1363310.
Building on the results of [13] and [12], we obtain optimal Suslin representations for mouse pairs. This leads us to an analysis of HOD , for any model M of ADR in which there are Wadge-cofinally many mouse pairs. We show also that if there is a least branch hod pair with a Woodin limit of Woodin cardinals, then there is a model of AD in which not every set of reals is ordinal definable from a countable sequence of ordinals.
σ−complete uniform ultrafilters where extensively studied in early seventies. Many nice results were obtained, specially by J. Ketonen (see [11]). Recently, G. Goldberg returned to the subject and proved a remarkable result that under The Ultrapower Axiom the first strongly compact cardinal is a supercompact. The purpose of the present paper is to provide some concrete examples of σ−complete uniform ultrafilters. 1 Some basic definitions and facts. Definition 1.1 Let U ⊆ P(λ) be an ultrafilter on λ. 1. U is called uniform iff for every A of cardinality < λ, λ \ A ∈ U . 2. U is called κ−complete iff the intersection of any less than κ members of U is in U . 3. U is called κ−complete exactly iff U is κ−complete, but not κ−complete. If κ is a strongly compact cardinal, then for every λ ≥ κ there is a uniform κ−complete ultrafilter over λ. By J. Ketonen [11] the opposite is true as well. Here we would like to examine the existence of σ−complete uniform ultrafilters over a cardinal λ which are exactly κ−complete for some κ < λ under much weaker assumptions. Note that if U is a σ−complete uniform ultrafilter over a cardinal λ and jU : V →MU ' Ult(V, U) is the corresponding elementary embedding, then sup(jU ′′λ) ≤ [id]U < jU(λ). Also, if U is κ−complete for some κ < λ, then cof(λ) ≥ κ. The following notion replaces the normality in the present context. ∗The work was partially supported by Israel Science Foundation Grants No. 58/14, 1216/18. We are grateful to the referee of the paper for his long list of suggestions, remarks and corrections.
The Posner-Robinson Theorem states that for any reals $Z$ and $A$ such that $Z \oplus 0' \leq_\mathrm{T} A$ and $0 <_\mathrm{T} Z$, there exists $B$ such that $A \equiv_\mathrm{T} B' \equiv_\mathrm{T} B \oplus Z \equiv_\mathrm{T} B \oplus 0'$. Consequently, any nonzero Turing degree $\operatorname{deg}_\mathrm{T}(Z)$ is a Turing jump relative to some $B$. Here we prove the hyperarithmetical analog, based on an unpublished proof of Slaman, namely that for any reals $Z$ and $A$ such that $Z \oplus \mathcal{O} \leq_\mathrm{T} A$ and $0 <_\mathrm{HYP} Z$, there exists $B$ such that $A \equiv_\mathrm{T} \mathcal{O}^B \equiv_\mathrm{T} B \oplus Z \equiv_\mathrm{T} B \oplus \mathcal{O}$. As an analogous consequence, any nonhyperarithmetical Turing degree $\operatorname{deg}_\mathrm{T}(Z)$ is a hyperjump relative to some $B$.
We investigate what collections of c.e. Turing degrees can be realised as the collection of elements of a separating Π^0_1 class of c.e. degree. We show that for every c.e. degree 𝐜, the collection {𝐜, 0'} can be thus realized. We also rule out several attempts at constructing separating classes realizing a unique c.e. degree. For example, we show that there is no super-maximal pair: disjoint c.e. sets A and B whose separating class is infinite, but every separator of c.e. degree is a finite variant of either A or B.
A function $F:2^\omega\to 2^\omega$ is an $E_0$-isomorphism if for all $x,y\in 2^\omega$, we have $xE_0y\iff f(x)E_0 f(y)$, where $xE_0y\iff(\exists a)(\forall n\ge b) x(n)=y(n)$. If such witnesses $a$ for $xE_0 y$ and for $f(x)E_0 f(y)$ depend on each other but not on $x$, $y$, then $F$ is called bi-uniform. It is shown that a homeomorphism of Cantor space which is a bi-uniform $E_0$-isomorphism can induce only the trivial automorphism of the Turing degrees.
We consider the reverse math strength of the statement $CDM$:``Every completely determined Borel set is measurable.'' Over $WWKL$, we obtain the following results analogous to the previously studied category case. $CDM$ lies strictly between $ATR$ and $L_{\omega_1,\omega}-CA$. Whenever $M\subseteq 2^\omega$ is the second-order part of an $\omega$-model of $CDM$, then for every $Z \in M$, there is a $R \in M$ such that $R$ is $\Delta^1_1$-random relative to $Z$. On the other hand, without $WWKL$, all sets have measure zero and thus $CDM$ loses its meaning. Vacuously, $\neg WWKL$ implies $CDM$ over $RCA$.
The functional interpretation is a systematic, syntactic method for transforming certain non-constructive proofs into constructive proofs with explicit bounds. We illustrate the interpretation by working through a concrete, fairly simple example, with almost no reference to formal logic, and then explain the connection with the underlying proof-theoretic methods.
This note is a proceeding of the workshop “On the Langlands Program: Endoscopy and Beyond” held in National University of Singapore from 17 Dec. 2018 to 18 Jan. 2019. The purpose is to explain Mœglin’s explicit constructions of A-packets both when the base field F is p-adic and when F is archimedean.
