scholarly journals HENKIN CONSTRUCTIONS OF MODELS WITH SIZE CONTINUUM

2019 ◽  
Vol 25 (1) ◽  
pp. 1-33 ◽  
Author(s):  
JOHN T. BALDWIN ◽  
MICHAEL C. LASKOWSKI
Keyword(s):  
Atomic Models ◽  
Perfect Set ◽  
Closure Relations

AbstractWe describe techniques for constructing models of size continuum inωsteps by simultaneously building a perfect set of enmeshed countable Henkin sets. Such models have perfect, asymptotically similar subsets. We survey applications involving Borel models, atomic models, two-cardinal transfers and models respecting various closure relations.

A COMBINATORIAL CORE OF THE ITERATED PERFECT SET MODEL

1999 ◽  
Vol 25 (1) ◽  
pp. 137
Author(s):  
Ciesielski
Keyword(s):  
Perfect Set

Constructible lattices of c-degrees

1982 ◽  
Vol 47 (4) ◽  
pp. 739-754
Author(s):  
C.P. Farrington
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Boolean Algebras ◽  
Combinatorial Complexity ◽  
Generic Extension ◽  
Transitive Model ◽  
Perfect Set ◽  
Consistency Results ◽  
Combinatorial Principles ◽  
Souslin Trees

This paper is devoted to the proof of the following theorem.Theorem. Let M be a countable standard transitive model of ZF + V = L, and let ℒ Є M be a wellfounded lattice in M, with top and bottom. Let ∣ℒ∣M = λ, and suppose κ ≥ λ is a regular cardinal in M. Then there is a generic extension N of M such that(i) N and M have the same cardinals, and κN ⊂ M;(ii) the c-degrees of sets of ordinals of N form a pattern isomorphic to ℒ;(iii) if A ⊂ On and A Є N, there is B Є P(κ+)N such that L(A) = L(B).The proof proceeds by forcing with Souslin trees, and relies heavily on techniques developed by Jech. In [5] he uses these techniques to construct simple Boolean algebras in L, and in [6] he uses them to construct a model of set theory whose c-degrees have orderlype 1 + ω*.The proof also draws on ideas of Adamovicz. In [1]–[3] she obtains consistency results concerning the possible patterns of c-degrees of sets of ordinals using perfect set forcing and symmetric models. These methods have the advantage of yielding real degrees, but involve greater combinatorial complexity, in particular the use of ‘sequential representations’ of lattices.The advantage of the approach using Souslin trees is twofold: first, we can make use of ready-made combinatorial principles which hold in L, and secondly, the notion of genericity over a Souslin tree is particularly simple.

scholarly journals Closure Relations fore±Pair Signatures in Gamma‐Ray Bursts

2008 ◽  
Vol 676 (2) ◽  
pp. 1123-1129 ◽  
Author(s):  
Kohta Murase ◽  
Kunihito Ioka
Keyword(s):  
Gamma Ray ◽  
Gamma Ray Bursts ◽  
Closure Relations

Sensitivity Analysis of Closure Relations for Numerical Prediction of Bubbly Flow in a Bubble Column.

2003 ◽  
Vol 29 (6) ◽  
pp. 778-786 ◽  
Author(s):  
Naoki Shimada ◽  
Akio Tomiyama ◽  
Iztok Zun ◽  
Hiroyuki Asano
Keyword(s):  
Sensitivity Analysis ◽  
Bubble Column ◽  
Bubbly Flow ◽  
Numerical Prediction ◽  
Closure Relations

scholarly journals THE NUMBER OF ATOMIC MODELS OF UNCOUNTABLE THEORIES

2018 ◽  
Vol 83 (1) ◽  
pp. 84-102
Author(s):  
DOUGLAS ULRICH
Keyword(s):  
Atomic Model ◽  
Complete Theory ◽  
Atomic Models ◽  
Image Position

AbstractWe show there exists a complete theory in a language of size continuum possessing a unique atomic model which is not constructible. We also show it is consistent with $ZFC + {\aleph _1} < {2^{{\aleph _0}}}$ that there is a complete theory in a language of size ${\aleph _1}$ possessing a unique atomic model which is not constructible. Finally we show it is consistent with $ZFC + {\aleph _1} < {2^{{\aleph _0}}}$ that for every complete theory T in a language of size ${\aleph _1}$, if T has uncountable atomic models but no constructible models, then T has ${2^{{\aleph _1}}}$ atomic models of size ${\aleph _1}$.

Atomic Models

1998 ◽  
pp. 269-285
Author(s):  
Narciso Garcia ◽  
Arthur Damask ◽  
Steven Schwarz
Keyword(s):  
Atomic Models
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