Comparing incomparable kleene degrees

1985 ◽  
Vol 50 (1) ◽  
pp. 55-58
Author(s):  
Philip Welch
Keyword(s):  
Turing Degrees ◽  
Generic Extension ◽  
Simple Application ◽  
Whole Process ◽  
Simple Sets ◽  
Image Position

In the December 1982 issue of this Journal Weitkamp [W] posed some questions concerning the incomparability of certain “r.e.” sets for the notion of Kleene reducibility. He asked whether the incomparability of, for example, the Friedman set F (defined below) and the set WI0 (the set of reals coding wellfounded trees of admissible height) was equivalent to the existence of 0#, since forcing over L with a set of conditions could not achieve this. We answer this by showing that in a certain class generic extension of L they are comparable, but 0# does not exist. This is an application of Jensen's coding theorem (cf. [BJW]), using a modified construction due to René David [D]. Indeed the result here is a simple application of his result. Define F as follows:Harrington showed, in effect, that one could not add a cone of Turing degrees to this set by forcing with sets of conditions over L. The method used here does add a cone of Turing degrees to a much simpler set RI1 (defined below)—and indeed the whole process could be viewed as forcing over L to obtain the determinacy of certain rather simple sets. It is the determinacy of the game with payoff set RI1 that ensures the comparability of F and WI0 (amongst many others).We shall refrain from repeating all the basic definitions and lemmas since the reader can readily refer to [W]; we shall give the basic necessities.

scholarly journals DEFINABLE MINIMAL COLLAPSE FUNCTIONS AT ARBITRARY PROJECTIVE LEVELS

2019 ◽  
Vol 84 (1) ◽  
pp. 266-289 ◽  
Author(s):  
VLADIMIR KANOVEI ◽  
VASSILY LYUBETSKY
Keyword(s):  
Generic Extension ◽  
Image Position

AbstractUsing a nonLaver modification of Uri Abraham’s minimal $\Delta _3^1$ collapse function, we define a generic extension $L[a]$ by a real a, in which, for a given $n \ge 3$, $\left\{ a \right\}$ is a lightface $\Pi _n^1 $ singleton, a effectively codes a cofinal map $\omega \to \omega _1^L $ minimal over L, while every $\Sigma _n^1 $ set $X \subseteq \omega $ is still constructible.

scholarly journals INDESTRUCTIBILITY OF THE TREE PROPERTY

2019 ◽  
Vol 85 (1) ◽  
pp. 467-485
Author(s):  
RADEK HONZIK ◽  
ŠÁRKA STEJSKALOVÁ
Keyword(s):  
Lower Bounds ◽  
Generic Extension ◽  
Tree Property ◽  
Weakly Compact ◽  
Cardinal Invariants ◽  
Cohen Forcing ◽  
Image Position

AbstractIn the first part of the article, we show that if $\omega \le \kappa < \lambda$ are cardinals, ${\kappa ^{ < \kappa }} = \kappa$, and λ is weakly compact, then in $V\left[M {\left( {\kappa ,\lambda } \right)} \right]$ the tree property at $$\lambda = \left( {\kappa ^{ + + } } \right)^{V\left[ {\left( {\kappa ,\lambda } \right)} \right]} $$ is indestructible under all ${\kappa ^ + }$-cc forcing notions which live in $V\left[ {{\rm{Add}}\left( {\kappa ,\lambda } \right)} \right]$, where ${\rm{Add}}\left( {\kappa ,\lambda } \right)$ is the Cohen forcing for adding λ-many subsets of κ and $\left( {\kappa ,\lambda } \right)$ is the standard Mitchell forcing for obtaining the tree property at $\lambda = \left( {\kappa ^{ + + } } \right)^{V\left[ {\left( {\kappa ,\lambda } \right)} \right]} $. This result has direct applications to Prikry-type forcing notions and generalized cardinal invariants. In the second part, we assume that λ is supercompact and generalize the construction and obtain a model ${V^{\rm{*}}}$, a generic extension of V, in which the tree property at ${\left( {{\kappa ^{ + + }}} \right)^{{V^{\rm{*}}}}}$ is indestructible under all ${\kappa ^ + }$-cc forcing notions living in $V\left[ {{\rm{Add}}\left( {\kappa ,\lambda } \right)} \right]$, and in addition under all forcing notions living in ${V^{\rm{*}}}$ which are ${\kappa ^ + }$-closed and “liftable” in a prescribed sense (such as ${\kappa ^{ + + }}$-directed closed forcings or well-met forcings which are ${\kappa ^{ + + }}$-closed with the greatest lower bounds).

scholarly journals THE COMPUTATIONAL CONTENT OF INTRINSIC DENSITY

2018 ◽  
Vol 83 (2) ◽  
pp. 817-828 ◽  
Author(s):  
ERIC P. ASTOR
Keyword(s):  
Reverse Mathematics ◽  
Previous Article ◽  
Asymptotic Density ◽  
Turing Degrees ◽  
Image Position ◽  
Immune Set

AbstractIn a previous article, the author introduced the idea of intrinsic density—a restriction of asymptotic density to sets whose density is invariant under computable permutation. We prove that sets with well-defined intrinsic density (and particularly intrinsic density 0) exist only in Turing degrees that are either high (${\bf{a}}\prime { \ge _{\rm{T}}}\emptyset \prime \prime$) or compute a diagonally noncomputable function. By contrast, a classic construction of an immune set in every noncomputable degree actually yields a set with intrinsic lower density 0 in every noncomputable degree.We also show that the former result holds in the sense of reverse mathematics, in that (over RCA0) the existence of a dominating or diagonally noncomputable function is equivalent to the existence of a set with intrinsic density 0.

scholarly journals Possible behaviours of the reflection ordering of stationary sets

1995 ◽  
Vol 60 (2) ◽  
pp. 534-547 ◽  
Author(s):  
Jiří Witzany
Keyword(s):  
Partial Ordering ◽  
Generic Extension ◽  
Maximal Antichain ◽  
Uncountable Cardinal ◽  
Image Position ◽  
Stationary Subsets ◽  
Almost All

AbstractIf S, T are stationary subsets of a regular uncountable cardinal κ, we say that S reflects fully in T, S < T, if for almost all α ∈ T (except a nonstationary set) S ∩ α stationary in α. This relation is known to be a well-founded partial ordering. We say that a given poset P is realized by the reflection ordering if there is a maximal antichain 〈Xp: p ∈ P〉 of stationary subsets of Reg(κ) so thatWe prove that if , and P is an arbitrary well-founded poset of cardinality ≤ κ+ then there is a generic extension where P is realized by the reflection ordering on κ.

scholarly journals COMPUTABLE STRUCTURES IN GENERIC EXTENSIONS

2016 ◽  
Vol 81 (3) ◽  
pp. 814-832 ◽  
Author(s):  
JULIA KNIGHT ◽  
ANTONIO MONTALBÁN ◽  
NOAH SCHWEBER
Keyword(s):  
Positive Result ◽  
Generic Extension ◽  
Ground Model ◽  
Rigid Structure ◽  
Basic Properties ◽  
Image Position ◽  
Scott Rank ◽  
The Universe ◽  
Generic Extensions ◽  
Countable Structure

AbstractIn this paper, we investigate connections between structures present in every generic extension of the universe V and computability theory. We introduce the notion of generic Muchnik reducibility that can be used to compare the complexity of uncountable structures; we establish basic properties of this reducibility, and study it in the context of generic presentability, the existence of a copy of the structure in every extension by a given forcing. We show that every forcing notion making ω2 countable generically presents some countable structure with no copy in the ground model; and that every structure generically presentable by a forcing notion that does not make ω2 countable has a copy in the ground model. We also show that any countable structure ${\cal A}$ that is generically presentable by a forcing notion not collapsing ω1 has a countable copy in V, as does any structure ${\cal B}$ generically Muchnik reducible to a structure ${\cal A}$ of cardinality ℵ1. The former positive result yields a new proof of Harrington’s result that counterexamples to Vaught’s conjecture have models of power ℵ1 with Scott rank arbitrarily high below ω2. Finally, we show that a rigid structure with copies in all generic extensions by a given forcing has a copy already in the ground model.

COMPUTABILITY IN UNCOUNTABLE BINARY TREES

2019 ◽  
Vol 84 (3) ◽  
pp. 1049-1098
Author(s):  
REESE JOHNSTON
Keyword(s):  
Binary Trees ◽  
Turing Degrees ◽  
Image Position ◽  
König’S Lemma

AbstractComputability, while usually performed within the context of ω, may be extended to larger ordinals by means of α-recursion. In this article, we concentrate on the particular case of ω1-recursion, and study the differences in the behavior of ${\rm{\Pi }}_1^0$-classes between this case and the standard one.Of particular interest are the ${\rm{\Pi }}_1^0$-classes corresponding to computable trees of countable width. Classically, it is well-known that the analog to König’s Lemma—“every tree of countable width and uncountable height has an uncountable branch”—fails; we demonstrate that not only does it fail effectively, but also that the failure is as drastic as possible. This is proven by showing that the ω1-Turing degrees of even isolated paths in computable trees of countable width are cofinal in the ${\rm{\Delta }}_1^1\,{\omega _1}$-Turing degrees.Finally, we consider questions of nonisolated paths, and demonstrate that the degrees realizable as isolated paths and the degrees realizable as nonisolated ones are very distinct; in particular, we show that there exists a computable tree of countable width so that every branch can only be ω1-Turing equivalent to branches of trees with ${\aleph _2}$-many branches.

scholarly journals THE COMPLEMENTS OF LOWER CONES OF DEGREES AND THE DEGREE SPECTRA OF STRUCTURES

2016 ◽  
Vol 81 (3) ◽  
pp. 997-1006 ◽  
Author(s):  
URI ANDREWS ◽  
MINGZHONG CAI ◽  
ISKANDER SH. KALIMULLIN ◽  
STEFFEN LEMPP ◽  
JOSEPH S. MILLER ◽  
...  
Keyword(s):  
Turing Degrees ◽  
Degree Spectra ◽  
Degree Spectrum ◽  
Image Position ◽  
Countable Structure

AbstractWe study Turing degrees a for which there is a countable structure ${\cal A}$ whose degree spectrum is the collection {x : x ≰ a}. In particular, for degrees a from the interval [0′, 0″], such a structure exists if a′ = 0″, and there are no such structures if a″ > 0‴.

scholarly journals On a conjecture of Dobrinen and Simpson concerning almost everywhere domination

2006 ◽  
Vol 71 (1) ◽  
pp. 119-136 ◽  
Author(s):  
Stephen Binns ◽  
Bjørn Kjos-Hanssen ◽  
Manuel Lerman ◽  
Reed Solomon
Keyword(s):  
Set Theory ◽  
Computable Function ◽  
Turing Degrees ◽  
Fair Coin ◽  
Almost Everywhere ◽  
Regularity Properties ◽  
Turing Reducibility ◽  
Image Position ◽  
Almost All ◽  
Generic Extensions

Dobrinen and Simpson [4] introduced the notions of almost everywhere domination and uniform almost everywhere domination to study recursion theoretic analogues of results in set theory concerning domination in generic extensions of transitive models of ZFC and to study regularity properties of the Lebesgue measure on 2ω in reverse mathematics. In this article, we examine one of their conjectures concerning these notions.Throughout this article, ≤T denotes Turing reducibility and μ denotes the Lebesgue (or “fair coin”) probability measure on 2ω given byA property holds almost everywhere or for almost all X ∈ 2ω if it holds on a set of measure 1. For f, g ∈ ωω, f dominatesg if ∃m∀n < m(f(n) > g(n)).(Dobrinen, Simpson). A set A ∈ 2ωis almost everywhere (a.e.) dominating if for almost all X ∈ 2ω and all functions g ≤TX, there is a function f ≤TA such that f dominates g. A is uniformly almost everywhere (u.a.e.) dominating if there is a function f ≤TA such that for almost all X ∈ 2ω and all functions g ≤TX, f dominates g.There are several trivial but useful observations to make about these definitions. First, although these properties are stated for sets, they are also properties of Turing degrees. That is, a set is (u.)a.e. dominating if and only if every other set of the same degree is (u.)a.e. dominating. Second, both properties are closed upwards in the Turing degrees. Third, u.a.e. domination implies a.e. domination. Finally, if A is u.a.e. dominating, then there is a function f ≤TA which dominates every computable function.

scholarly journals COARSE REDUCIBILITY AND ALGORITHMIC RANDOMNESS

2016 ◽  
Vol 81 (3) ◽  
pp. 1028-1046 ◽  
Author(s):  
DENIS R. HIRSCHFELDT ◽  
CARL G. JOCKUSCH ◽  
RUTGER KUYPER ◽  
PAUL E. SCHUPP
Keyword(s):  
Main Tool ◽  
The Other ◽  
Compactness Theorem ◽  
Random Sets ◽  
Symmetric Difference ◽  
Turing Degrees ◽  
Algorithmic Randomness ◽  
Random Set ◽  
Natural Embedding ◽  
Image Position

AbstractA coarse description of a set A ⊆ ω is a set D ⊆ ω such that the symmetric difference of A and D has asymptotic density 0. We study the extent to which noncomputable information can be effectively recovered from all coarse descriptions of a given set A, especially when A is effectively random in some sense. We show that if A is 1-random and B is computable from every coarse description D of A, then B is K-trivial, which implies that if A is in fact weakly 2-random then B is computable. Our main tool is a kind of compactness theorem for cone-avoiding descriptions, which also allows us to prove the same result for 1-genericity in place of weak 2-randomness. In the other direction, we show that if $A \le _{{\rm{T}}} \emptyset {\rm{'}}$ is a 1-random set, then there is a noncomputable c.e. set computable from every coarse description of A, but that not all K-trivial sets are computable from every coarse description of some 1-random set. We study both uniform and nonuniform notions of coarse reducibility. A set Y is uniformly coarsely reducible to X if there is a Turing functional Φ such that if D is a coarse description of X, then ΦD is a coarse description of Y. A set B is nonuniformly coarsely reducible to A if every coarse description of A computes a coarse description of B. We show that a certain natural embedding of the Turing degrees into the coarse degrees (both uniform and nonuniform) is not surjective. We also show that if two sets are mutually weakly 3-random, then their coarse degrees form a minimal pair, in both the uniform and nonuniform cases, but that the same is not true of every pair of relatively 2-random sets, at least in the nonuniform coarse degrees.

scholarly journals ON THE INTERPLAY BETWEEN EFFECTIVE NOTIONS OF RANDOMNESS AND GENERICITY

2019 ◽  
Vol 84 (1) ◽  
pp. 393-407
Author(s):  
LAURENT BIENVENU ◽  
CHRISTOPHER P. PORTER
Keyword(s):  
Random Sequence ◽  
Complete Classification ◽  
Minimal Pair ◽  
Turing Degrees ◽  
Random Oracles ◽  
Image Position

AbstractIn this paper, we study the power and limitations of computing effectively generic sequences using effectively random oracles. Previously, it was known that every 2-random sequence computes a 1-generic sequence (as shown by Kautz) and every 2-random sequence forms a minimal pair in the Turing degrees with every 2-generic sequence (as shown by Nies, Stephan, and Terwijn). We strengthen these results by showing that every Demuth random sequence computes a 1-generic sequence and that every Demuth random sequence forms a minimal pair with every pb-generic sequence (where pb-genericity is an effective notion of genericity that is strictly between 1-genericity and 2-genericity). Moreover, we prove that for every comeager${\cal G} \subseteq {2^\omega }$, there is some weakly 2-random sequenceXthat computes some$Y \in {\cal G}$, a result that allows us to provide a fairly complete classification as to how various notions of effective randomness interact in the Turing degrees with various notions of effective genericity.

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