MARGINALIA ON A THEOREM OF WOODIN

2017 ◽  
Vol 82 (1) ◽  
pp. 359-374
Author(s):  
RASMUS BLANCK ◽  
ALI ENAYAT
Keyword(s):  
Fixed Point ◽  
Peano Arithmetic ◽  
Countable Model ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Finite Set ◽  
Image Position

AbstractLet $\left\langle {{W_n}:n \in \omega } \right\rangle$ be a canonical enumeration of recursively enumerable sets, and suppose T is a recursively enumerable extension of PA (Peano Arithmetic) in the same language. Woodin (2011) showed that there exists an index $e \in \omega$ (that depends on T) with the property that if${\cal M}$ is a countable model of T and for some${\cal M}$-finite set s, ${\cal M}$ satisfies ${W_e} \subseteq s$, then${\cal M}$ has an end extension${\cal N}$ that satisfies T + We = s.Here we generalize Woodin’s theorem to all recursively enumerable extensions T of the fragment ${{\rm{I}\rm{\Sigma }}_1}$ of PA, and remove the countability restriction on ${\cal M}$ when T extends PA. We also derive model-theoretic consequences of a classic fixed-point construction of Kripke (1962) and compare them with Woodin’s theorem.

Partial degrees and the density problem. Part 2: The enumeration degrees of the Σ2 sets are dense

1984 ◽  
Vol 49 (2) ◽  
pp. 503-513 ◽  
Author(s):  
S. B. Cooper
Keyword(s):  
Density Theorem ◽  
Turing Degrees ◽  
Partial Functions ◽  
Enumeration Degrees ◽  
Recursively Enumerable ◽  
Recursive Sequences ◽  
Recursively Enumerable Sets ◽  
Finite Set ◽  
Image Position ◽  
Density Problem

As in Rogers [3], we treat the partial degrees as notational variants of the enumeration degrees (that is, the partial degree of a function is identified with the enumeration degree of its graph). We showed in [1] that there are no minimal partial degrees. The purpose of this paper is to show that the partial degrees below 0′ (that is, the partial degrees of the Σ2 partial functions) are dense. From this we see that the Σ2 sets play an analagous role within the enumeration degrees to that played by the recursively enumerable sets within the Turing degrees. The techniques, of course, are very different to those required to prove the Sacks Density Theorem (see [4, p. 20]) for the recursively enumerable Turing degrees.Notation and terminology are similar to those of [1]. In particular, We, Dx, 〈m, n〉, ψe are, respectively, notations for the e th r.e. set in a given standard listing of the r.e. sets, the finite set whose canonical index is x, the recursive code for (m, n) and the e th enumeration operator (derived from We). Recursive approximations etc. are also defined as in [1].Theorem 1. If B and C are Σ2sets of numbers, and B ≰e C, then there is an e-operator Θ withProof. We enumerate an e-operator Θ so as to satisfy the list of conditions:Let {Bs ∣ s ≥ 0}, {Cs ∣ s ≥ 0} be recursive sequences of approximations to B, C respectively, for which, for each х, х ∈ B ⇔ (∃s*)(∀s ≥ s*)(х ∈ Bs) and х ∈ C ⇔ (∃s*)(∀s ≥ s*)(х ∈ Cs).

scholarly journals Interpretability over peano arithmetic

1999 ◽  
Vol 64 (4) ◽  
pp. 1407-1425
Author(s):  
Claes Strannegård
Keyword(s):  
Modal Logic ◽  
Completeness Theorem ◽  
Peano Arithmetic ◽  
Embedding Theorem ◽  
Compactness Theorem ◽  
Partial Answer ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Interpretability Logic

AbstractWe investigate the modal logic of interpretability over Peano arithmetic. Our main result is a compactness theorem that extends the arithmetical completeness theorem for the interpretability logic ILMω. This extension concerns recursively enumerable sets of formulas of interpretability logic (rather than single formulas). As corollaries we obtain a uniform arithmetical completeness theorem for the interpretability logic ILM and a partial answer to a question of Orey from 1961. After some simplifications, we also obtain Shavrukov's embedding theorem for Magari algebras (a.k.a. diagonalizable algebras).

scholarly journals ENAYAT MODELS OF PEANO ARITHMETIC

2018 ◽  
Vol 83 (04) ◽  
pp. 1501-1511 ◽  
Author(s):  
ATHAR ABDUL-QUADER
Keyword(s):  
Linear Order ◽  
Peano Arithmetic ◽  
Countable Model ◽  
Natural Question ◽  
Conservative Extension ◽  
Image Position ◽  
Definable Elements

AbstractSimpson [6] showed that every countable model ${\cal M} \models PA$ has an expansion $\left( {{\cal M},X} \right) \models P{A^{\rm{*}}}$ that is pointwise definable. A natural question is whether, in general, one can obtain expansions of a nonprime model in which the definable elements coincide with those of the underlying model. Enayat [1] showed that this is impossible by proving that there is ${\cal M} \models PA$ such that for each undefinable class X of ${\cal M}$, the expansion $\left( {{\cal M},X} \right)$ is pointwise definable. We call models with this property Enayat models. In this article, we study Enayat models and show that a model of $PA$ is Enayat if it is countable, has no proper cofinal submodels and is a conservative extension of all of its elementary cuts. We then show that, for any countable linear order γ, if there is a model ${\cal M}$ such that $Lt\left( {\cal M} \right) \cong \gamma$, then there is an Enayat model ${\cal M}$ such that $Lt\left( {\cal M} \right) \cong \gamma$.

Π10 classes and Boolean combinations of recursively enumerable sets

1974 ◽  
Vol 39 (1) ◽  
pp. 95-96 ◽  
Author(s):  
Carl G. Jockusch
Keyword(s):  
Turing Machine ◽  
Peano Arithmetic ◽  
Binary Sequences ◽  
First Order ◽  
Recursive Tree ◽  
Recursively Enumerable ◽  
Private Correspondence ◽  
Recursively Enumerable Sets ◽  
Binary Input

Let be the collection of all sets which are finite Boolean combinations of recursively enumerable (r.e.) sets of numbers. Dale Myers asked [private correspondence] whether there exists a nonempty class of sets containing no member of . In this note we construct such a class. The motivation for Myers' question was his observation (reported in [1]) that the existence of such a class is equivalent to the assertion that there is a finite consistent set of tiles which has no m-trial tiling of the plane for any m (obeying the “origin constraint”). (For explanations of these terms and further results on tilings of the plane, cf. [1] and [5].) In addition to the application to tilings, the proof of our results gives some information on bi-immune sets and on complete extensions of first-order Peano arithmetic.A class of sets may be roughly described as the class of infinite binary input tapes for which a fixed Turing machine fails to halt, or alternatively as the class of infinite branches of a recursive tree of finite binary sequences. (In these definitions, sets of numbers are identified with the corresponding binary sequences.) Precise definitions, as well as many results concerning such classes, may be found in [3] and [4].

Arithmetic and the theory of types

1984 ◽  
Vol 49 (2) ◽  
pp. 621-624 ◽  
Author(s):  
M. Boffa
Keyword(s):  
Peano Arithmetic ◽  
Positive Answer ◽  
Second Order ◽  
Countable Model ◽  
Natural Numbers ◽  
Conservative Extension ◽  
Logical Definition ◽  
Claude Bernard ◽  
Definition Of ◽  
Image Position

A hundred years ago, Frege proposed a logical definition of the natural numbers based on the following idea:He replaced this circular definition by the following one:He tried afterwards to found his theory over a notion of class satisfying a general comprehension principle:Russell quickly derived a contradiction from this principle (the famous Russell's paradox) but saved Frege's arithmetic with his theory of types based on the following comprehension principle:In 1979, talking at the Claude Bernard University in Lyon, I remarked that 3 types suffice to provide Frege's arithmetic, showing in fact that PA2 (second order Peano arithmetic) holds in TT3 + AI (theory of types 0, 1, 2 plus a suitable axiom of infinity). I asked whether TT3 + AI was a conservative extension of PA2. Pabion [3] gave a positive answer by a subtle use of the Fraenkel-Moskowski method. This result will be improved in the present paper, with a view to getting models of NF3 + AI in which Frege's arithmetic forms a model isomorphic to a given countable model of PA2.

A hierarchy of families of recursively enumerable degrees

1984 ◽  
Vol 49 (4) ◽  
pp. 1160-1170 ◽  
Author(s):  
Lawrence V. Welch
Keyword(s):  
Finite Sets ◽  
Recursively Enumerable ◽  
Similar Theorem ◽  
Recursively Enumerable Sets ◽  
Complete Set ◽  
Image Position

Certain investigations have been made concerning the nature of classes of recursively enumerable sets, and the relation of such classes to the recursively enumerable indices of their sets. For instance, a theorem of Rice [3, Theorem XIV(a), p. 324] states that if A is the complete set of indices for a class of recursively enumerable sets (that is, if there is a class of recursively enumerable sets such that and if A is recursive, then either A = ⌀ or A = ω. A relate theorem by Rice and Shapiro [3, Theorem XIV(b), p. 324] can be stated as follows:Let be a class of recursively enumerable sets, and let A be the complete set of indices for . Then A is r.e. if and only if there is an r.e. set D of canonical indices of finite sets Du, u ∈ D, such thatA somewhat similar theorem of Yates is the following: Let be a class of recursively enumerable sets which contains all finite sets. Let A be the complete set of indices for . Then there is a uniform recursive enumeration of the sets in if and only if A is recursively enumerable in 0(2)—that is, if and only if A is Σ3. A corollary of this is that if C is any r.e. set such that C(2)≡T⌀(2), there is a uniform recursive enumeration of all sets We such that We ≤TC [9, Theorem 9, p. 265].

Some representations of Diophantine sets

1972 ◽  
Vol 37 (3) ◽  
pp. 572-578 ◽  
Author(s):  
Raphael M. Robinson
Keyword(s):  
Universal Quantifier ◽  
Natural Numbers ◽  
Recursively Enumerable Set ◽  
Minor Error ◽  
Integer Coefficients ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Image Position ◽  
A Minor

A set D of natural numbers is called Diophantine if it can be defined in the formwhere P is a polynomial with integer coefficients. Recently, Ju. V. Matijasevič [2], [3] has shown that all recursively enumerable sets are Diophantine. From this, it follows that a bound for n may be given.We use throughout the logical symbols ∧ (and), ∨ (or), → (if … then …), ↔ (if and only if), ⋀ (for every), and ⋁ (there exists); negation does not occur explicitly. The variables range over the natural numbers 0,1,2,3, …, except as otherwise noted.It is the purpose of this paper to show that if we do not insist on prenex form, then every Diophantine set can be defined existentially by a formula in which not more than five existential quantifiers are nested. Besides existential quantifiers, only conjunctions are needed. By Matijasevič [2], [3], the representation extends to all recursively enumerable sets. Using this, we can find a bound for the number of conjuncts needed.Davis [1] proved that every recursively enumerable set of natural numbers can be represented in the formwhere P is a polynomial with integer coefficients. I showed in [5] that we can take λ = 4. (A minor error is corrected in an Appendix to this paper.) By the methods of the present paper, we can again obtain this result, and indeed in a stronger form, with the universal quantifier replaced by a conjunction.

scholarly journals AXIOMATIZATION OF PROVABLE n-PROVABILITY

2019 ◽  
Vol 84 (02) ◽  
pp. 849-869 ◽  
Author(s):  
EVGENY KOLMAKOV ◽  
LEV BEKLEMISHEV
Keyword(s):  
Peano Arithmetic ◽  
Additional Information ◽  
Recursively Enumerable ◽  
Speed Up ◽  
Reflection Principles ◽  
Image Position

AbstractA formula φ is called n-provable in a formal arithmetical theory S if φ is provable in S together with all true arithmetical ${{\rm{\Pi }}_n}$-sentences taken as additional axioms. While in general the set of all n-provable formulas, for a fixed $n > 0$, is not recursively enumerable, the set of formulas φ whose n-provability is provable in a given r.e. metatheory T is r.e. This set is deductively closed and will be, in general, an extension of S. We prove that these theories can be naturally axiomatized in terms of progressions of iterated local reflection principles. In particular, the set of provably 1-provable sentences of Peano arithmetic $PA$ can be axiomatized by ${\varepsilon _0}$ times iterated local reflection schema over $PA$. Our characterizations yield additional information on the proof-theoretic strength of these theories (w.r.t. various measures of it) and on their axiomatizability. We also study the question of speed-up of proofs and show that in some cases a proof of n-provability of a sentence can be much shorter than its proof from iterated reflection principles.

A boundedness theorem in ID1(W)

1986 ◽  
Vol 51 (4) ◽  
pp. 942-947
Author(s):  
Gerhard Jäger
Keyword(s):  
Fixed Point ◽  
Peano Arithmetic ◽  
Boundedness Theorem ◽  
Recursive Tree ◽  
Free Variables ◽  
Positive Formula ◽  
Image Position ◽  
Recursive Trees

In this paper we prove a boundedness theorem in the theory ID1(W). This answers a question asked by Feferman, for example in [3]. The background is the following.Let A[X, x] be an X-positive formula arithmetic in X. The theory ID1(PA) is an extension of Peano arithmetic PA by the following axioms:for arbitrary formulas F; PA is a constant for the least fixed point of A[X, x]. Set-theoretically, PA can be defined by recursion on the ordinals as follows:where is the first nonrecursive ordinal.Now let a ≺ b be the arithmetic relation which expresses that the recursive tree coded by a is a proper subtree of the tree coded by b, and defineThe least fixed point of Tree[X, x] is the set PTree of all well-founded recursive trees. We write W or Wα for PTree or , respectively. Since W is complete we have for all α < . If we define for each element a ∈ W its inductive norm ∣a∣ by ∣a∣≔ min{ξ: a ∈ Wξ}, then we have = {∣a∣: a ∈ W} and the elements of W can be used as codes for the ordinals less than .Assume that B[X, x] is an X-positive formula arithmetic in X with the only free variables X and x, and assume that QB is a relation that satisfiesIf we definethen we obviously have PB = IB.

Recursively enumerable sets modulo iterated jumps and extensions of Arslanov's completeness criterion

1989 ◽  
Vol 54 (4) ◽  
pp. 1288-1323 ◽  
Author(s):  
C. G. Jockusch ◽  
M. Lerman ◽  
R. I. Soare ◽  
R. M. Solovay
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Equivalence Relations ◽  
Main Concern ◽  
Strict Sense ◽  
Recursively Enumerable ◽  
Completeness Criterion ◽  
Starting Point ◽  
Recursively Enumerable Sets ◽  
Point Form

Let We be the eth recursively enumerable (r.e.) set in a standard enumeration. The fixed point form of Kleene's recursion theorem asserts that for every recursive function f there exists e which is a fixed point of f in the sense that We = Wf(e). In this paper our main concern is to study the degrees of functions with no fixed points. We consider both fixed points in the strict sense above and fixed points modulo various equivalence relations on recursively enumerable sets.Our starting point for the investigation of the degrees of functions without (strict) fixed points is the following result due to M. M. Arslanov [A1, Theorem 1] and known as the Arslanov completeness criterion. Proofs of this result may also be found in [So1, Theorem 1.3] and [So2, Chapter 12], and we will give a game version of the proof in §5 of this paper.Theorem 1.1 (Arslanov). Let A be an r.e. set. Then A is complete (i.e. A has degree0′) iff there is a function f recursive in A with no fixed point.

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