Almost local non-α-recursiveness

1974 ◽  
Vol 39 (3) ◽  
pp. 552-562 ◽  
Author(s):  
Chi T. Chong
Keyword(s):  
Initial Segment ◽  
Continuum Hypothesis ◽  
General Question ◽  
Reflection Property ◽  
Recursively Enumerable ◽  
Generalized Continuum ◽  
Background Material ◽  
Admissible Ordinals ◽  
Recursive Set

§1. Let α be an admissible ordinal which is also a limit of admissible ordinals (e.g. take any α such that α = α*, its projectum [5]). For any admissible γ ≦ α, let [γ) denote the initial segment of ordinals less than γ. A very general question that one might ask is the following: What conditions should one put on γ so that a certain statement true in Lα is ‘reflected’ to be true in Lγ? We cite some examples: (a) If β < γ < α, then Lα ⊨ “β cardinal” is ‘reflected’ to Lγ ⊨ “ β cardinal” (⊨ is just the satisfaction relation). (b) If β < γ < α and γ is a cardinal in Lα (called α-cardinal for short), then Lα ⊨ “β is not a cardinal” is ‘reflected’ to Lγ ⊨ “β is not a cardinal” This fact is used in Gödel's proof that V = L implies the Generalized Continuum Hypothesis. Our objective in this paper is to study a ‘reflection’ property of the following sort: Let A ⊆ [α) be an α-recursively enumerable (α-r.e.), non-α-recursive set. Under what conditions will A restricted to a smaller admissible ordinal γ be γ-r.e. and not γ-recursive?The notations used here are standard. Those that are not explained are adopted from the paper of Sacks and Simpson [5], to which we also refer the reader for background material.

An α-finite injury method of the unbounded type

1976 ◽  
Vol 41 (1) ◽  
pp. 1-17
Author(s):  
C. T. Chong
Keyword(s):  
Fine Structure ◽  
Initial Segment ◽  
Fundamental Importance ◽  
Upper Semilattice ◽  
The Past ◽  
Recursively Enumerable ◽  
Background Material ◽  
A Priority ◽  
Admissible Ordinals ◽  
Priority Argument

Let α be an admissible ordinal. In this paper we study the structure of the upper semilattice of α-recursively enumerable degrees. Various results about the structure which are of fundamental importance had been obtained during the past two years (Sacks-Simpson [7], Lerman [4], Shore [9]). In particular, the method of finite priority argument of Friedberg and Muchnik was successfully generalized in [7] to an α-finite priority argument to give a solution of Post's problem for all admissible ordinals. We refer the reader to [7] for background material, and we also follow closely the notations used there.Whereas [7] and [4] study priority arguments in which the number of injuries inflicted on a proper initial segment of requirements can be effectively bounded (Lemma 2.3 of [7]), we tackle here priority arguments in which no such bounds exist. To this end, we focus our attention on the fine structure of Lα, much in the fashion of Jensen [2], and show that we can still use a priority argument on an indexing set of requirements just short enough to give us the necessary bounds we seek.

The uniform regular set theorem in α-recursion theory

1978 ◽  
Vol 43 (2) ◽  
pp. 270-279 ◽  
Author(s):  
Wolfgang Maass
Keyword(s):  
Initial Segment ◽  
Recursion Theory ◽  
Typical Situation ◽  
Recursively Enumerable ◽  
Constructible Sets ◽  
Admissible Ordinals ◽  
Special Case

Several new features arise in the generalization of recursion theory on ω to recursion theory on admissible ordinals α, thus making α-recursion theory an interesting theory. One of these is the appearance of irregular sets. A subset A of α is called regular (over α), if we have for all β < α that A ∩ B ∈ Lα, otherwise A is called irregular (over α). So in the special case of ordinary recursion theory (α = ω) every subset of α is regular, but if α is not a cardinal of L we find constructible sets A ⊆ α which are irregular. The notion of regularity becomes essential, if we deal with α-recursively enumerable (α-r.e.) sets in priority constructions (α-r.e. is defined as Σ1 over Lα). The typical situation occurring there is that an α-r.e. set A is enumerated during some construction in which one tries to satisfy certain requirements. Often this construction succeeds only if we can insure that every initial segment A ∩ β of A is completely enumerated at some stage before α. This calls for making sure that A is regular because due to the admissibility of α an α-r.e. set A is regular iff for every (or equivalently for one) enumeration f of A (f is an enumeration of A iff f: α → A is α-recursive, total, 1-1 and onto) we have that is the image of the set σ under f).

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.

scholarly journals Isomorphic Universality and the Number of Pairwise Nonisomorphic Models in the Class of Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Mirna Džamonja
Keyword(s):  
Banach Spaces ◽  
Continuum Hypothesis ◽  
Generalized Continuum

We develop the framework ofnatural spacesto study isomorphic embeddings of Banach spaces. We then use it to show that a sufficient failure of the generalized continuum hypothesis implies that the universality number of Banach spaces of a given density under a certain kind of positive embedding (very positive embedding) is high. An example of a very positive embedding is a positive onto embedding betweenC(K)andCLfor 0-dimensionalKandLsuch that the following requirement holds for allh≠0andf≥0inC(K): if0≤Th≤Tf, then there are constantsa≠0andbwith0≤a·h+b≤fanda·h+b≠0.

Ultraproducts which are not saturated

1967 ◽  
Vol 32 (1) ◽  
pp. 23-46 ◽  
Author(s):  
H. Jerome Keisler
Keyword(s):  
Predicate Logic ◽  
Continuum Hypothesis ◽  
First Order ◽  
Generalized Continuum ◽  
First Order Predicate Logic

In this paper we continue our study, begun in [5], of the connection between ultraproducts and saturated structures. IfDis an ultrafilter over a setI, andis a structure (i.e., a model for a first order predicate logicℒ), the ultrapower ofmoduloDis denoted byD-prod. The ultrapower is important because it is a method of constructing structures which are elementarily equivalent to a given structure(see Frayne-Morel-Scott [3]). Our ultimate aim is to find out what kinds of structure are ultrapowers of. We made a beginning in [5] by proving that, assuming the generalized continuum hypothesis (GCH), for each cardinalαthere is an ultrafilterDover a set of powerαsuch that for all structures,D-prodisα+-saturated.

On the inadequacy of inner models

1972 ◽  
Vol 37 (3) ◽  
pp. 569-571
Author(s):  
Andreas Blass
Keyword(s):  
Standard Model ◽  
Continuum Hypothesis ◽  
Axiom Of Choice ◽  
Precise Statement ◽  
Relative Consistency ◽  
Inner Models ◽  
Generalized Continuum ◽  
The Axiom Of Choice

The method of inner models, used by Gödel to prove the (relative) consistency of the axiom of choice and the generalized continuum hypothesis [2], cannot be used to prove the (relative) consistency of any statement which contradicts the axiom of constructibility (V = L). A more precise statement of this well-known fact is:(*)For any formula θ(x) of the language of ZF, there is an axiom α of the theory ZF + V ≠ L such that the relativization α(θ) is not a theorem of ZF.On p. 108 of [1], Cohen gives a proof of (*) in ZF assuming the existence of a standard model of ZF, and he indicates that this assumption can be avoided. However, (*) is not a theorem of ZF (unless ZF is inconsistent), because (*) trivially implies the consistency of ZF. What assumptions are needed to prove (*)? We know that the existence of a standard model implies (*) which, in turn, implies the consistency of ZF. Is either implication reversible?From our main result, it will follow that, if the converse of the first implication is provable in ZF, then ZF has no standard model, and if the converse of the second implication is provable in ZF, then so is the inconsistency of ZF. Thus, it is quite improbable that either converse is provable in ZF.

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