Andreas Blass and Saharon Shelah. Ultrafilters with small generating sets. Israel journal of mathematics, vol. 65 (1989), pp. 259–271. - Andreas Blass and Saharon Shelah. There may be simple - and -points and the Rudin–Keisler ordering may be downward directed. Annals of pure and applied logic, vol. 33 (1987), pp. 213–243. - Andreas Blass. Near coherence of filters. II: Applications to operator ideals, the Stone–Čech remainder of a half-line, order ideals of sequences, and the slenderness of groups. Transactions of the American Mathematical Society, vol. 300 (1987), pp. 557–581. - Andreas Blass and Saharon Shelah. Near coherence of filters III: a simplified consistency proof. Notre Dame journal of formal logic, vol. 30 (1989), pp. 530–538. - Andreas Blass and Claude Laflamme. Consistency results about filters and the number of inequivalent growth types. The journal of symbolic logic, vol. 54 (1989), pp. 50–56. - Andreas Blass. Applications of superperfect forcing and its relatives. Set theory and its applications. Proceedings of a conference held at York University, Ontario, Canada, Aug. 10–21, 1987, edited by J. Steprāns and S. Watson, Lecture notes in mathematics, vol. 1401, Springer-Verlag, Berlin etc. 1989, pp. 18–40. - Andreas Blass and Saharon Shelah. Ultrafilters with small generating sets. Israel journal of mathematics, vol. 65 (1989), pp. 259–271.

1992 ◽  
Vol 57 (2) ◽  
pp. 763-766
Author(s):  
Peter J. Nyikos
Keyword(s):  
Set Theory ◽  
Israel Journal ◽  
American Mathematical Society ◽  
Operator Ideals ◽  
Consistency Proof ◽  
Applied Logic ◽  
Generating Sets ◽  
Consistency Results ◽  
Growth Types ◽  
Order Ideals

Arthur W. Apter. On the least strongly compact cardinal. Israel journal of mathematics, vol. 35 (1980), pp. 225–233. - Arthur W. Apter. Measurability and degrees of strong compactness. The journal of symbolic logic, vol. 46 (1981), pp. 249–254. - Arthur W. Apter. A note on strong compactness and supercompactness. Bulletin of the London Mathematical Society, vol. 23 (1991), pp. 113–115. - Arthur W. Apter. On the first n strongly compact cardinals. Proceedings of the American Mathematical Society, vol. 123 (1995), pp. 2229–2235. - Arthur W. Apter and Saharon Shelah. On the strong equality between supercompactness and strong compactness.. Transactions of the American Mathematical Society, vol. 349 (1997), pp. 103–128. - Arthur W. Apter and Saharon Shelah. Menas' result is best possible. Ibid., pp. 2007–2034. - Arthur W. Apter. More on the least strongly compact cardinal. Mathematical logic quarterly, vol. 43 (1997), pp. 427–430. - Arthur W. Apter. Laver indestructibility and the class of compact cardinals. The journal of symbolic logic, vol. 63 (1998), pp. 149–157. - Arthur W. Apter. Patterns of compact cardinals. Annals of pure and applied logic, vol. 89 (1997), pp. 101–115. - Arthur W. Apter and Moti Gitik. The least measurable can be strongly compact and indestructible. The journal of symbolic logic, vol. 63 (1998), pp. 1404–1412.

2000 ◽  
Vol 6 (1) ◽  
pp. 86-89
Author(s):  
James W. Cummings
Keyword(s):  
Israel Journal ◽  
American Mathematical Society ◽  
London Mathematical Society ◽  
Mathematical Society ◽  
Symbolic Logic ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Applied Logic ◽  
Strong Compactness ◽  
Strongly Compact Cardinals

Consistency results about filters and the number of inequivalent growth types

1989 ◽  
Vol 54 (1) ◽  
pp. 50-56 ◽  
Author(s):  
Andreas Blass ◽  
Claude Laflamme
Keyword(s):  
Set Theory ◽  
Partial Ordering ◽  
Consistency Results ◽  
Growth Types ◽  
Element Chain ◽  
Combinatorial Principles ◽  
Proper Filter ◽  
Positive Integers ◽  
Nondecreasing Functions ◽  
Models Of Set Theory

We use models of set theory described in [2] and [3] to prove the consistency of several combinatorial principles, for example:If ℱ is any filter on N containing all the cofinite sets, then there is a finite-to-one function f: N → N such that f(ℱ) is either the filter of cofinite sets or an ultrafilter.As a consequence of our combinatorial principles, we also obtain the consistency of:The partial ordering P of slenderness classes of abelian groups, denned and studied in [4], is a four-element chain.In the remainder of this Introduction, we shall define our terminology and state the combinatorial principles to be considered. In §2, we shall establish some implications between these principles. In §3, we shall prove our consistency results by showing that the strongest of our principles holds in models of set theory constructed in [2] and [3].A filter on N will always mean a proper filter containing all cofinite sets; in particular, an ultrafilter will necessarily be nonprincipal. We write N ↗ N for the set of nondecreasing functions from the set N of positive integers into itself. A subset ℐ of N ↗ N is called an ideal if it is closed downward (if f(n) ≤ g(n) for all n and if g ∈ ℐ, then f ∈ ℐ) and closed under binary maximum (if f(n) = max(g(n), h(n)) for all n and if g, h ∈ ℐ then f ∈ ℐ).

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 Forcing with Non-wellfounded Models

2007 ◽  
Vol 5 ◽  
Author(s):  
Paul Corazza
Keyword(s):  
Set Theory ◽  
Ground Model ◽  
Iterated Forcing ◽  
Consistency Results

We develop the machinery for performing forcing over an arbitrary (possibly non-wellfounded) model of set theory. For consistency results, this machinery is unnecessary since such results can always be legitimately obtained by assuming that the ground model is (countable) transitive. However, for establishing properties of a given (possibly non-wellfounded) model, the fully developed machinery of forcing as a means to produce new related models can be useful. We develop forcing through iterated forcing, paralleling the standard steps of presentation found in [19] and [14].

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