Stepping up lemmas in definable partitions

1984 ◽  
Vol 49 (1) ◽  
pp. 22-31 ◽  
Author(s):  
Evangelos Kranakis
Keyword(s):  
Partition Relations ◽  
Admissible Ordinals

AbstractSeveral stepping up lemmas are proved which are then used to investigate the connection between definable partition relations and admissible ordinals.

NEGATIVE PARTITION RELATIONS OR UNCOUNTABLE CARDINALS OF COFINALITY

2001 ◽  
Vol 37 (1-2) ◽  
pp. 233-236
Author(s):  
P. Matet
Keyword(s):  
Partition Relations ◽  
Uncountable Cardinals

We modify an argument of Baumgartner to show that…

A lift of a theorem of Friedberg: A Banach-Mazur functional that coincides with no α-recursive functional on the class of α-recursive functions

1981 ◽  
Vol 46 (2) ◽  
pp. 216-232 ◽  
Author(s):  
Robert A. di Paola
Keyword(s):  
Recursive Functions ◽  
Admissible Ordinals

AbstractR. M. Friedberg demonstrated the existence of a recursive functional that agrees with no Banach-Mazur functional on the class of recursive functions. In this paper Friedberg's result is generalized to both α-recursive functionals and weak α-recursive functionals for all admissible ordinals α such that λ < α*, where α* is the Σ1-projectum of α and λ is the Σ2-cofinality of α. The theorem is also established for the metarecursive case, α = ω1, where α* = λ = ω.

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.

Partition Relations

2009 ◽  
pp. 129-213 ◽  
Author(s):  
András Hajnal ◽  
Jean A. Larson
Keyword(s):  
Partition Relations

Model theory under the axiom of determinateness

1985 ◽  
Vol 50 (3) ◽  
pp. 773-780
Author(s):  
Mitchell Spector
Keyword(s):  
Model Theory ◽  
Axiom Of Choice ◽  
Order Logic ◽  
First Order Logic ◽  
Technical Innovation ◽  
First Order ◽  
Partition Relations ◽  
Hanf Number ◽  
The Axiom Of Choice

AbstractWe initiate the study of model theory in the absence of the Axiom of Choice, using the Axiom of Determinateness as a powerful substitute. We first show that, in this context, is no more powerful than first-order logic. The emphasis then turns to upward Löwenhein-Skolem theorems; ℵ1 is the Hanf number of first-order logic, of , and of a strong fragment of , The main technical innovation is the development of iterated ultrapowers using infinite supports; this requires an application of infinite-exponent partition relations. All our theorems can be proven from hypotheses weaker than AD.

On Finite Polarized Partition Relations

1969 ◽  
Vol 12 (3) ◽  
pp. 321-326 ◽  
Author(s):  
V. Chvátal
Keyword(s):  
Cardinal Numbers ◽  
Partition Relations ◽  
Image Position

Call an m × n array an m × n; k array if its mn entries come from a set of k elements. An m × n; 1 array has mn like entries. We write(1)if every m × n; k array contains a p × q; 1 sub-array. The negation of (1) is writtenand means that there is an m × n; k array containing no p × q; 1 sub-array. Relations (1) are called "polarized partition relations among cardinal numbers" by P. Erdös and R. Rado [2]. In this note we prove the following theorems.

scholarly journals FORCING AND THE HALPERN–LÄUCHLI THEOREM

2019 ◽  
Vol 85 (1) ◽  
pp. 87-102
Author(s):  
NATASHA DOBRINEN ◽  
DANIEL HATHAWAY
Keyword(s):  
Partition Relations ◽  
Finite Products

AbstractWe investigate the effects of various forcings on several forms of the Halpern– Läuchli theorem. For inaccessible κ, we show they are preserved by forcings of size less than κ. Combining this with work of Zhang in [17] yields that the polarized partition relations associated with finite products of the κ-rationals are preserved by all forcings of size less than κ over models satisfying the Halpern– Läuchli theorem at κ. We also show that the Halpern–Läuchli theorem is preserved by <κ-closed forcings assuming κ is measurable, following some observed reflection properties.

scholarly journals Partition relations for successor cardinals

1986 ◽  
Vol 59 (2) ◽  
pp. 152-169 ◽  
Author(s):  
Akihiro Kanamori
Keyword(s):  
Partition Relations
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