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.

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).

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.

α-degrees of α-theories

1972 ◽  
Vol 37 (4) ◽  
pp. 677-682 ◽  
Author(s):  
George Metakides
Keyword(s):  
Initial Segment ◽  
Recursion Theory ◽  
The Other ◽  
Infinite Length ◽  
Infinitary Logic ◽  
Admissible Set ◽  
Recursively Enumerable ◽  
Limit Ordinal ◽  
The Subject ◽  
Further Development

Let α be a limit ordinal with the property that any “recursive” function whose domain is a proper initial segment of α has its range bounded by α. α is then called admissible (in a sense to be made precise later) and a recursion theory can be developed on it (α-recursion theory) by providing the generalized notions of α-recursively enumerable, α-recursive and α-finite. Takeuti [12] was the first to study recursive functions of ordinals, the subject owing its further development to Kripke [7], Platek [8], Kreisel [6], and Sacks [9].Infinitary logic on the other hand (i.e., the study of languages which allow expressions of infinite length) was quite extensively studied by Scott [11], Tarski, Kreisel, Karp [5] and others. Kreisel suggested in the late '50's that these languages (even which allows countable expressions but only finite quantification) were too large and that one should only allow expressions which are, in some generalized sense, finite. This made the application of generalized recursion theory to the logic of infinitary languages appear natural. In 1967 Barwise [1] was the first to present a complete formalization of the restriction of to an admissible fragment (A a countable admissible set) and to prove that completeness and compactness hold for it. [2] is an excellent reference for a detailed exposition of admissible languages.

Σ-predicates of finite types over an admissible set

1985 ◽  
Vol 24 (5) ◽  
pp. 327-351 ◽  
Author(s):  
Yu. L. Ershov
Keyword(s):  
Admissible Set

Uncountable models and infinitary elementary extensions

1973 ◽  
Vol 38 (3) ◽  
pp. 460-470 ◽  
Author(s):  
John Gregory
Keyword(s):  
Countable Subset ◽  
Elementary Extension ◽  
Admissible Set ◽  
Natural Numbers ◽  
Elementary Equivalence ◽  
Countable Structure

Let A be a countable admissible set (as defined in [1], [3]). The language LA consists of all infinitary finite-quantifier formulas (identified with sets, as in [1]) that are elements of A. Notationally, LA = A ∩ Lω1ω. Then LA is a countable subset of Lω1ω, the language of all infinitary finite-quantifier formulas with all conjunctions countable. The set is the set of Lω1ω sentences defined in 2.2 below. The following theorem characterizes those A-Σ1 sets Φ of LA sentences that have uncountable models.Main Theorem (3.1.). If Φ is an A-Σ1set of LA sentences, then the following are equivalent:(a) Φ has an uncountable model,(b) Φ has a model with a proper LA-elementary extension,(c) for every , ⋀Φ → C is not valid.This theorem was announced in [2] and is proved in §§3, 4, 5. Makkai's earlier [4, Theorem 1] implies that, if Φ determines countable structure up to Lω1ω-elementary equivalence, then (a) is equivalent to (c′) for all , ⋀Φ → C is not valid.The requirement in 3.1 that Φ is A-Σ1 is essential when the set ω of all natural numbers is an element of A. For by the example of [2], then there is a set Φ LA sentences such that (b) holds and (a) fails; it is easier to show that, if ω ϵ A, there is a set Φ of LA sentences such that (c) holds and (b) fails.

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.

A note on the undefinability of cuts

1983 ◽  
Vol 48 (3) ◽  
pp. 564-569 ◽  
Author(s):  
J.B. Paris ◽  
C. Dimitracopoulos
Keyword(s):  
First Order ◽  
Nonstandard Model ◽  
Image Position

The results in this paper were motivated by the following result due to R. Solovay.Theorem 1 (Solovay). Let M be a nonstandard model of Peano's first order axioms P and let I ⊂e M (i.e. ϕ ≠ ⊂ M and I is closed under < and successor). Then for each of the functions we can define J ⊆e I in ‹M, I› such that J is closed under that function. (∣x∣ denotes [log2(x)].)Proof. Just notice that the cuts defined byare successively closed under In view of Theorem 1, the following question was raised by R. Solovay: Can we define J ⊆ I in ‹M, I› such that J is closed under exponentiation? In Theorem 2 we show that the answer is “no”. Theorem 3 is based on Theorem 2 and extends the technique to cuts which are models of subsystems of P.To prove both theorems we shall need an estimate due to R. Parikh (see [1], especially the proof of Theorem 2.2a). For the sake of completeness, and also to introduce some notation we shall sketch Parikh's estimate in the next section. At all times we shall give the easiest estimates which still work rather than the sharpest ones.

NX Reuse Library Workflow Based on Knowledge

2014 ◽  
Vol 687-691 ◽  
pp. 2521-2524
Author(s):  
Xiang Hui Zhan ◽  
Xiao Da Li
Keyword(s):  
Management System ◽  
Important Application ◽  
Part Family ◽  
Standard Parts Library ◽  
Standard Part ◽  
Parts Library ◽  
Reuse Library

In reuse library platform, realizing the reuse process of standard parts and common parts and building standard part management system are the important application during the whole design field. This paper introduces the method of customization standard parts library using part family, and realizes standardization management of reuse library. By creating KRX files, the knowledge components in reuse library are managed. Reuse library is an effective way to organize and use standard parts and common parts.

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