Locally countable models of Σ1-separation

1981 ◽  
Vol 46 (1) ◽  
pp. 96-100 ◽  
Author(s):  
Fred G. Abramson
Keyword(s):  
Nonstandard Model ◽  
Standard Part ◽  
Locally Countable ◽  
Transitive Set ◽  
Countable Models

AbstractLet α be any countable admissible ordinal greater than ω. There is a transitive set A such that A is admissible, locally countable, OnA = α, and A satisfies Σ1-separation. In fact, if B is any nonstandard model of KP + ∀x ⊆ ω (the hyperjump of x exists), the ordinal standard part of B is greater than ω, and every standard ordinal in B is countable in B, then HCB ∩ (standard part of B) satisfies Σ-separation.

Σ1-separation

1979 ◽  
Vol 44 (3) ◽  
pp. 374-382
Author(s):  
Fred G. Abramson
Keyword(s):  
Admissible Set ◽  
Nonstandard Model ◽  
Standard Part

AbstractLet A be a standard transitive admissible set. Σ1-separation is the principle that whenever X and Y are disjoint Σ1A subsets of A then there is a ⊿1A subset S of A such that X ⊆ S and Y ∩ S = ∅.Theorem. If satisfies Σ-separation, then(1) If 〈Tn∣n < ω) ϵ A is a sequence of trees on ω each of which has at most finitely many infinite paths in A then the function n ↦ (set of infinite paths in A through Tn) is in A.(2) If A is not closed under hyperjump and α = OnA then A has in it a nonstandard model of V = L whose ordinal standard part is α.Theorem. Let α be any countable admissible ordinal greater than ω. Then there is a model of Σ1-separation whose height is α.

Two theorems on degrees of models of true arithmetic

1984 ◽  
Vol 49 (2) ◽  
pp. 425-436 ◽  
Author(s):  
Julia Knight ◽  
Alistair H. Lachlan ◽  
Robert I. Soare
Keyword(s):  
Standard Model ◽  
Binary Operation ◽  
Peano Arithmetic ◽  
Turing Degree ◽  
First Order ◽  
Nonstandard Model ◽  
Basis Theorem ◽  
The Standard Model ◽  
The Universe ◽  
Countable Models

Let PA be the theory of first order Peano arithmetic, in the language L with binary operation symbols + and ·. Let N be the theory of the standard model of PA. We consider countable models M of PA such that the universe ∣M∣ is ω. The degree of such a model M, denoted by deg(M), is the (Turing) degree of the atomic diagram of M. The results of this paper concern the degrees of models of N, but here in the Introduction, we shall give a brief survey of results about degrees of models of PA.Let D0 denote the set of degrees d such that there is a nonstandard model of M of PA with deg(M) = d. Here are some of the more easily stated results about D0.(1) There is no recursive nonstandard model of PA; i.e., 0 ∈ D0.This is a result of Tennenbaum [T].(2) There existsd ∈ D0such thatd ≤ 0′.This follows from the standard Henkin argument.(3) There existsd ∈ D0such thatd < 0′.Shoenfield [Sh1] proved this, using the Kreisel-Shoenfield basis theorem.(4) There existsd ∈ D0such thatd′ = 0′.Jockusch and Soare [JS] improved the Kreisel-Shoenfield basis theorem and obtained (4).(5) D0 = Dc = De, where Dc denotes the set of degrees of completions of PA and De the set of degrees d such that d separates a pair of effectively inseparable r.e. sets.Solovay noted (5) in a letter to Soare in which in answer to a question posed in [JS] he showed that Dc is upward closed.

Flipping properties in arithmetic

1982 ◽  
Vol 47 (2) ◽  
pp. 416-422 ◽  
Author(s):  
L. A. S. Kirby
Keyword(s):  
Standard Model ◽  
Set Theory ◽  
Initial Segment ◽  
Peano Arithmetic ◽  
Combinatorial Properties ◽  
Nonstandard Model ◽  
Standard Part ◽  
The Standard Model ◽  
Nonstandard Models ◽  
Image Position

Flipping properties were introduced in set theory by Abramson, Harrington, Kleinberg and Zwicker [1]. Here we consider them in the context of arithmetic and link them with combinatorial properties of initial segments of nonstandard models studied in [3]. As a corollary we obtain independence resutls involving flipping properties.We follow the notation of the author and Paris in [3] and [2], and assume some knowledge of [3]. M will denote a countable nonstandard model of P (Peano arithmetic) and I will be a proper initial segment of M. We denote by N the standard model or the standard part of M. X ↑ I will mean that X is unbounded in I. If X ⊆ M is coded in M and M ≺ K, let X(K) be the subset of K coded in K by the element which codes X in M. So X(K) ⋂ M = X.Recall that M ≺IK (K is an I-extension of M) if M ≺ K and for some c∈K,In [3] regular and strong initial segments are defined, and among other things it is shown that I is regular if and only if there exists an I-extension of M.

On the standard part of nonstandard models of set theory

1983 ◽  
Vol 48 (1) ◽  
pp. 33-38 ◽  
Author(s):  
Menachem Magidor ◽  
Saharon Shelah ◽  
Jonathan Stavi
Keyword(s):  
Set Theory ◽  
Nonstandard Model ◽  
Standard Part ◽  
Nonstandard Models ◽  
Models Of Set Theory

AbstractWe characterize the ordinals α of uncountable cofinality such that α is the standard part of a nonstandard model of ZFC (or equivalently KP).

scholarly journals EVERY COUNTABLE MODEL OF SET THEORY EMBEDS INTO ITS OWN CONSTRUCTIBLE UNIVERSE

2013 ◽  
Vol 13 (02) ◽  
pp. 1350006 ◽  
Author(s):  
JOEL DAVID HAMKINS
Keyword(s):  
Set Theory ◽  
Order Type ◽  
Large Cardinals ◽  
Countable Model ◽  
Classical Theorem ◽  
Binary Relations ◽  
Finite Sets ◽  
Nonstandard Model ◽  
Models Of Set Theory ◽  
Countable Models

The main theorem of this article is that every countable model of set theory 〈M, ∈M〉, including every well-founded model, is isomorphic to a submodel of its own constructible universe 〈LM, ∈M〉 by means of an embedding j : M → LM. It follows from the proof that the countable models of set theory are linearly pre-ordered by embeddability: if 〈M, ∈M〉 and 〈N, ∈N〉 are countable models of set theory, then either M is isomorphic to a submodel of N or conversely. Indeed, these models are pre-well-ordered by embeddability in order-type exactly ω1 + 1. Specifically, the countable well-founded models are ordered under embeddability exactly in accordance with the heights of their ordinals; every shorter model embeds into every taller model; every model of set theory M is universal for all countable well-founded binary relations of rank at most Ord M; and every ill-founded model of set theory is universal for all countable acyclic binary relations. Finally, strengthening a classical theorem of Ressayre, the proof method shows that if M is any nonstandard model of PA, then every countable model of set theory — in particular, every model of ZFC plus large cardinals — is isomorphic to a submodel of the hereditarily finite sets 〈 HF M, ∈M〉 of M. Indeed, 〈 HF M, ∈M〉 is universal for all countable acyclic binary relations.

Using binary operations to construct a transitive set of block transformations

2020 ◽  
Vol 30 (6) ◽  
pp. 375-389
Author(s):  
Igor V. Cherednik
Keyword(s):  
Binary Operation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Binary Operations ◽  
The Family ◽  
Transitive Set ◽  
Necessary And Sufficient

AbstractWe study the set of transformations {ΣF : F∈ 𝓑∗(Ω)} implemented by a network Σ with a single binary operation F, where 𝓑∗(Ω) is the set of all binary operations on Ω that are invertible as function of the second variable. We state a criterion of bijectivity of all transformations from the family {ΣF : F∈ 𝓑∗(Ω)} in terms of the structure of the network Σ, identify necessary and sufficient conditions of transitivity of the set of transformations {ΣF : F∈ 𝓑∗(Ω)}, and propose an efficient way of verifying these conditions. We also describe an algorithm for construction of networks Σ with transitive sets of transformations {ΣF : F∈ 𝓑∗(Ω)}.

SINGULAR CYCLES AND Ck-ROBUST TRANSITIVE SET ON MANIFOLD WITH BOUNDARY

2011 ◽  
Vol 13 (02) ◽  
pp. 191-211 ◽  
Author(s):  
D. CARRASCO-OLIVERA ◽  
C. A. MORALES ◽  
B. SAN MARTÍN
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Manifold With Boundary ◽  
Transitive Set ◽  
Singular Cycle ◽  
Hyperbolic Equilibrium

Let M be a 3-manifold with boundary ∂M. Let X be a C∞, vector field on M, tangent to ∂M, exhibiting a singular cycle associated to a hyperbolic equilibrium σ∈∂M with real eigenvalues λss < λs < 0 < λu satisfying λs - λss - 2λu > 0. We prove under generic conditions and k large enough the existence of a Ck robust transitive set of X, that is, any Ck vector field Ck close to X exhibits a transitive set containing the cycle. In particular, C∞ vector fields exhibiting Ck robust transitive sets, for k large enough, which are not singular-hyperbolic do exist on any compact 3-manifold with boundary.

scholarly journals ULTRAHIGH ENERGY NEUTRINOS

2003 ◽  
Vol 18 (22) ◽  
pp. 4085-4096 ◽  
Author(s):  
SHARADA IYER DUTTA ◽  
MARY HALL RENO ◽  
INA SARCEVIC
Keyword(s):  
Black Hole ◽  
Standard Model ◽  
Cross Section ◽  
Neutrino Oscillations ◽  
Ultrahigh Energy ◽  
Nonstandard Model ◽  
Air Shower ◽  
Energy Neutrino ◽  
The Standard Model ◽  
Event Rates

The ultrahigh energy neutrino cross section is well understood in the standard model for neutrino energies up to 1012 GeV, Tests of neutrino oscillations (νμ ↔ ντ) from extragalactic sources of neutrinos are possible with large underground detectors. Measurements of horizontal air shower event rates at neutrino energies above 1010 GeV will be able to constrain nonstandard model contributions to the neutrino-nucleon cross section, e.g., from mini-black hole production.

Remarks on weak notions of saturation in models of Peano arithmetic

1987 ◽  
Vol 52 (1) ◽  
pp. 129-148 ◽  
Author(s):  
Matt Kaufmann ◽  
James H. Schmerl
Keyword(s):  
Peano Arithmetic ◽  
Saturated Model ◽  
Simple Extension ◽  
Minimal Models ◽  
Recursively Saturated ◽  
Countable Models

This paper is a sequel to our earlier paper [2] entitled Saturation and simple extensions of models of Peano arithmetic. Among other things, we will answer some of the questions that were left open there. In §1 we consider the question of whether there are lofty models of PA which have no recursively saturated, simple extensions. We are still unable to answer this question; but we do show in that section that these models are precisely the lofty models which are not recursively saturated and which are κ-like for some regular κ. In §2 we use diagonal methods to produce minimal models of PA in which the standard cut is recursively definable, and other minimal models in which the standard cut is not recursively definable. In §3 we answer two questions from [2] by exhibiting countable models of PA which, in the terminology of this paper, are uniformly ω-lofty but not continuously ω-lofty and others which are continuously ω-lofty but not recursively saturated. We also construct a model (assuming ◇) which is not recursively saturated but every proper, simple cofinal extension of which is ℵ1-saturated. Finally, in §4 we answer another question from [2] by proving that for regular κ ≥ ℵ1; every κ-saturated model of PA has a κ-saturated proper, simple extension which is not κ+-saturated.Our notation and terminology are quite standard. Anything unfamiliar to the reader and not adequately denned here is probably defined in §1 of [2]. All models considered are models of Peano arithmetic.

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