Some results on higher Suslin trees

1990 ◽  
Vol 55 (2) ◽  
pp. 526-536 ◽  
Author(s):  
R. David
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Suslin Tree

Higher Suslin trees have become a tool in some forcing constructions in set theory (see, for example, [D1] and [D2]). Most of the constructions using ω1 Suslin trees can be extended to κ+ Suslin trees for any regular cardinal κ. Some of these are given in §1.In many such constructions, sequences of Suslin trees are used. In §II we show, in various ways, that the generalization to sequences, even ω-sequences, of κ+ Suslin trees cannot be done.In these constructions the Suslin trees are used as forcing poset (the forcing adds a branch in the tree). There is another way to kill a Suslin tree, namely by adding a big antichain. Some results on this forcing are given in §III.Our notation is standard. If T is a tree and x ∈ T, then ∣x∣ is the height of x in T. We define Tα (or T(α)) = {x ∈ T: ∣x∣ = α} and T∣α = {x ∈ T: ∣x∣ < α}.If p and q are forcing conditions, p ≤ q means that p has more information than q.If (Tα: α ∈ I) is a sequence of trees, Π Tα will always mean the set of (xα: α ∈ I) such that xα ∈ Tα and ∣xα∣ = ∣xβ∣ for α, β ∈ I.For functions b, T,… we denote by b ∣α, T∣ α,… their restriction to α.If x is a sequence of ordinals and α is an ordinal, x∧α is the sequence obtained by concatenating α at the end of x.

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 Almost-Free E-Rings of Cardinality ℵ1

2003 ◽  
Vol 55 (4) ◽  
pp. 750-765
Author(s):  
Rüdiger Göbel ◽  
Saharon Shelah ◽  
Lutz Strüngmann
Keyword(s):  
Abelian Group ◽  
Set Theory ◽  
Regular Cardinal ◽  
Additive Structure ◽  
Unital Ring ◽  
Ring Element ◽  
Models Of Set Theory

AbstractAn E-ring is a unital ring R such that every endomorphism of the underlying abelian group R+ is multiplication by some ring element. The existence of almost-free E-rings of cardinality greater than 2ℵ0 is undecidable in ZFC. While they exist in Gödel's universe, they do not exist in other models of set theory. For a regular cardinal ℵ1 ≤ λ 2ℵ0 we construct E-rings of cardinality λ in ZFC which have ℵ1-free additive structure. For λ = ℵ1 we therefore obtain the existence of almost-free E-rings of cardinality ℵ1 in ZFC.

Blunt and topless end extensions of models of set theory

1983 ◽  
Vol 48 (4) ◽  
pp. 1053-1073 ◽  
Author(s):  
Matt Kaufmann
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Models Of Set Theory ◽  
Satisfaction Class

AbstractLet be a well-founded model of ZFC whose class of ordinals has uncountable cofinality, such that has a Σn end extension for each n ∈ ω. It is shown in Theorem 1.1 that there is such a model which has no elementary end extension. In the process some interesting facts about topless end extensions (those with no least new ordinal) are uncovered, for example Theorem 2.1: If is a well-founded model of ZFC, such that has uncountable cofinality and has a topless Σ3 end extension, then has a topless elementary end extension and also a well-founded elementary end extension, and contains ordinals which are (in ) highly hyperinaccessible. In §3 related results are proved for κ-like models (κ any regular cardinal) which need not be well founded. As an application a soft proof is given of a theorem of Schmerl on the model-theoretic relation κ → λ. (The author has been informed that Silver had earlier, independently, found a similar unpublished proof of that theorem.) Also, a simpler proof is given of (a generalization of) a characterization by Keisler and Silver of the class of well-founded models which have a Σn end extension for each n ∈ ω. The case κ = ω1 is investigated more deeply in §4, where the problem solved by Theorem 1.1 is considered for non-well-founded models. In Theorems 4.1 and 4.4, ω1-like models of ZFC are constructed which have a Σn end extension for all n ∈ ω but have no elementary end extension. ω1-like models of ZFC which have no Σ3 end extension are produced in Theorem 4.2. The proof uses a notion of satisfaction class, which is also applied in the proof of Theorem 4.6: No model of ZFC has a definable end extension which satisfies ZFC. Finally, Theorem 5.1 generalizes results of Keisler and Morley, and Hutchinson, by asserting that every model of ZFC of countable cofinality has a topless elementary end extension. This contrasts with the rest of the paper, which shows that for well-founded models of uncountable cofinality and for κ-like models with κ regular, topless end extensions are much rarer than blunt end extensions.

Borel reductions and cub games in generalised descriptive set theory

2013 ◽  
Vol 78 (2) ◽  
pp. 439-458 ◽  
Author(s):  
Vadim Kulikov
Keyword(s):  
Partial Order ◽  
Set Theory ◽  
Regular Cardinal ◽  
Equivalence Relations ◽  
Descriptive Set Theory ◽  
Borel Equivalence Relations ◽  
Power Set ◽  
Borel Reducibility ◽  
Descriptive Set

AbstractIt is shown that the power set of κ ordered by the subset relation modulo various versions of the non-stationary ideal can be embedded into the partial order of Borel equivalence relations on 2κ under Borel reducibility. Here κ is an uncountable regular cardinal with κ<κ = κ.

Elementary extensions of countable models of set theory

1976 ◽  
Vol 41 (1) ◽  
pp. 139-145 ◽  
Author(s):  
John E. Hutchinson
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Countable Model ◽  
Elementary Extension ◽  
Models Of Set Theory ◽  
Countable Models

AbstractWe prove the following extension of a result of Keisler and Morley. Suppose is a countable model of ZFC and c is an uncountable regular cardinal in . Then there exists an elementary extension of which fixes all ordinals below c, enlarges c, and either (i) contains or (ii) does not contain a least new ordinal.Related results are discussed.

The consistency of the axiom of universality for the ordering of cardinalities

1985 ◽  
Vol 50 (2) ◽  
pp. 502-509
Author(s):  
Marco Forti ◽  
Furio Honsell
Keyword(s):  
Set Theory ◽  
Partially Ordered Set ◽  
Regular Cardinal ◽  
Ordered Set ◽  
Partial Ordering ◽  
Generic Extension ◽  
Ground Model ◽  
Partially Ordered ◽  
Transfer Method ◽  
Image Position

T. Jech [4] and M. Takahashi [7] proved that given any partial ordering R in a model of ZFC there is a symmetric submodel of a generic extension of where R is isomorphic to the injective ordering on a set of cardinals.The authors raised the question whether the injective ordering of cardinals can be universal, i.e. whether the following axiom of “cardinal universality” is consistent:CU. For any partially ordered set (X, ≼) there is a bijection f:X → Y such that(i.e. x ≼ y iff ∃g: f(x) → f(y) injective). (See [1].)The consistency of CU relative to ZF0 (Zermelo-Fraenkel set theory without foundation) is proved in [2], but the transfer method of Jech-Sochor-Pincus cannot be applied to obtain consistency with full ZF (including foundation), since CU apparently is not boundable.In this paper the authors define a model of ZF + CU as a symmetric submodel of a generic extension obtained by forcing “à la Easton” with a class of conditions which add κ generic subsets to any regular cardinal κ of a ground model satisfying ZF + V = L.

Trees and -subsets of ω1ω1

1993 ◽  
Vol 58 (3) ◽  
pp. 1052-1070 ◽  
Author(s):  
Alan Mekler ◽  
Jouko Väänänen
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Type Theorem ◽  
Descriptive Set Theory ◽  
Covering Property ◽  
Borel Set ◽  
Similar Role ◽  
Universal Family ◽  
Definable Subset ◽  
Descriptive Set

AbstractWe study descriptive set theory in the space by letting trees with no uncountable branches play a similar role as countable ordinals in traditional descriptive set theory. By using such trees, we get, for example, a covering property for the class of -sets of .We call a family of trees universal for a class of trees if ⊆ and every tree in can be order-preservingly mapped into a tree in . It is well known that the class of countable trees with no infinite branches has a universal family of size ℵ1. We shall study the smallest cardinality of a universal family for the class of trees of cardinality ≤ ℵ1 with no uncountable branches. We prove that this cardinality can be 1 (under ¬CH) and any regular cardinal κ which satisfies (under CH). This bears immediately on the covering property of the -subsets of the space .We also study the possible cardinalities of definable subsets of . We show that the statement that every definable subset of has cardinality <ωn or cardinality is equiconsistent with ZFC (if n ≥ 3) and with ZFC plus an inaccessible (if n = 2).Finally, we define an analogue of the notion of a Borel set for the space and prove a Souslin-Kleene type theorem for this notion.

scholarly journals Internal Consistency and the Inner Model Hypothesis

2006 ◽  
Vol 12 (4) ◽  
pp. 591-600 ◽  
Author(s):  
Sy-David Friedman
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Weak Sense ◽  
Large Cardinals ◽  
Model Method ◽  
Inner Models ◽  
Inner Model ◽  
The Relationship ◽  
The Universe ◽  
The Way

There are two standard ways to establish consistency in set theory. One is to prove consistency using inner models, in the way that Gödel proved the consistency of GCH using the inner model L. The other is to prove consistency using outer models, in the way that Cohen proved the consistency of the negation of CH by enlarging L to a forcing extension L[G].But we can demand more from the outer model method, and we illustrate this by examining Easton's strengthening of Cohen's result:Theorem 1 (Easton's Theorem). There is a forcing extensionL[G] of L in which GCH fails at every regular cardinal.Assume that the universe V of all sets is rich in the sense that it contains inner models with large cardinals. Then what is the relationship between Easton's model L[G] and V? In particular, are these models compatible, in the sense that they are inner models of a common third model? If not, then the failure of GCH at every regular cardinal is consistent only in a weak sense, as it can only hold in universes which are incompatible with the universe of all sets. Ideally, we would like L[G] to not only be compatible with V, but to be an inner model of V.We say that a statement is internally consistent iff it holds in some inner model, under the assumption that there are innermodels with large cardinals.

Set theory with a filter quantifier

1983 ◽  
Vol 48 (2) ◽  
pp. 263-287 ◽  
Author(s):  
Matt Kaufmann
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Completeness Theorem ◽  
Large Cardinal ◽  
Second Order ◽  
Class Theory ◽  
Natural Extensions ◽  
Definition Of ◽  
Almost All

The incompleteness of ZF set theory leads one to look for natural extensions of ZF in which one can prove statements independent of ZF which appear to be “true”. One approach has been to add large cardinal axioms. Or, one can investigate second-order expansions like Kelley-Morse class theory, KM. In this paper we look at a set theory ZF(aa), with an added quantifier aa which ranges over ordinals. The “aa” stands for “almost all”, and although we will consider interpretations in terms of the closed unbounded filter on a regular cardinal κ, we will consider other interpretations also.We start in §1 by giving the axioms for the theory ZF(aa) and presenting a completeness theorem which gives a model-theoretic definition of ZF(aa). In §2 we investigate set theory with a satisfaction predicate and interpret it in a fragment of ZF(aa). In §3 we generalize the methods of §2 to obtain a hierarchy of satisfaction predicates. We use these predicates to prove reflection theorems, as well as to prove the consistency of certain fragments of ZF(aa). Next, in §4 we discuss expandability of models of ZF to models of fragments of ZF(aa) and of Kelley-Morse. We conclude in §5 with a discussion of an extension ZF(aa) + DET of ZF(aa) in which the quantifier aa is self-dual.

Sign in / Sign up

Export Citation Format

Share Document