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.

Boolean extensions which efface the Mahlo property

1974 ◽  
Vol 39 (2) ◽  
pp. 254-268 ◽  
Author(s):  
William Boos
Keyword(s):  
Set Theory ◽  
Large Cardinals ◽  
Large Cardinal ◽  
Boolean Algebras ◽  
Partial Coverage ◽  
Transitive Model ◽  
Consistency Results ◽  
Lower Case ◽  
Basic Notation ◽  
Generic Extensions

The results that follow are intended to be understood as informal counterparts to formal theorems of Zermelo-Fraenkel set theory with choice. Basic notation not explained here can usually be found in [5]. It will also be necessary to assume a knowledge of the fundamentals of boolean and generic extensions, in the style of Jech's monograph [3]. Consistency results will be stated as assertions about the existence of certain complete boolean algebras, B, C, etc., either outright or in the sense of a countable standard transitive model M of ZFC augmented by hypotheses about the existence of various large cardinals. Proofs will usually be phrased in terms of the forcing relation ⊩ over such an M, especially when they make heavy use of genericity. They are then assertions about Shoenfield-style P-generic extensions M(G), in which the ‘names’ are required without loss of generality to be elements of MB = (VB)M, B is the boolean completion of P in M (cf. [3, p. 50]: the notation there is RO(P)), the generic G is named by Ĝ ∈ MB such that (⟦p ∈ Ĝ⟧B = p and (cf. [11, p. 361] and [3, pp. 58–59]), and for p ∈ P and c1, …, cn ∈ MB, p ⊩ φ(c1, …, cn) iff ⟦φ(c1, …, cn)⟧B ≥ p (cf. [3, pp. 61–62]).Some prior acquaintance with large cardinal theory is also needed. At this writing no comprehensive introductory survey is yet in print, though [1], [10], [12]and [13] provide partial coverage. The scheme of definitions which follows is intended to fix notation and serve as a glossary for reference, and it is followed in turn by a description of the results of the paper. We adopt the convention that κ, λ, μ, ν, ρ and σ vary over infinite cardinals, and all other lower case Greek letters (except χ, φ, ψ, ϵ) over arbitrary ordinals.

On generic extensions without the axiom of choice

1983 ◽  
Vol 48 (1) ◽  
pp. 39-52 ◽  
Author(s):  
G. P. Monro
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Generic Extension ◽  
Transitive Model ◽  
The Axiom Of Choice ◽  
Generic Extensions

AbstractLet ZF denote Zermelo-Fraenkel set theory (without the axiom of choice), and let M be a countable transitive model of ZF. The method of forcing extends M to another model M[G] of ZF (a “generic extension”). If the axiom of choice holds in M it also holds in M[G], that is, the axiom of choice is preserved by generic extensions. We show that this is not true for many weak forms of the axiom of choice, and we derive an application to Boolean toposes.

scholarly journals Outer models and genericity

2003 ◽  
Vol 68 (2) ◽  
pp. 389-418 ◽  
Author(s):  
M. C. Stanley
Keyword(s):  
Set Theory ◽  
Generic Extension ◽  
Transitive Model ◽  
Class Forcing ◽  
The Universe ◽  
Models Of Set Theory

Why is forcing the only known method for constructing outer models of set theory?If V is a standard transitive model of ZFC, then a standard transitive model W of ZFC is an outer model of V if V ⊆ W and V ∩ OR = W ∩ OR.Is every outer model of a given model a generic extension? At one point Solovay conjectured that if 0# exists, then every real that does not construct 0# lies in L[G], for some G that is generic for some forcing ℙ ∈ L. Famously, this was refuted by Jensen's coding theorem. He produced a real that is generic for an L-definable class forcing property, but does not lie in any set forcing extension of L.Beller, Jensen, and Welch in Coding the universe [BJW] revived Solovay's conjecture by asking the following question: Let a ⊆ ω be such that L[a] ⊨ “0# does not exist”. Is there ab∈ L[a] such that a ∉ L[b] and a is set generic over L[b].

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).

SQUARE WITH BUILT-IN DIAMOND-PLUS

2017 ◽  
Vol 82 (3) ◽  
pp. 809-833 ◽  
Author(s):  
ASSAF RINOT ◽  
RALF SCHINDLER
Keyword(s):  
Regular Cardinal ◽  
Infinite Cardinal ◽  
List Type ◽  
Type Number ◽  
Aronszajn Tree ◽  
Combinatorial Principles ◽  
Souslin Trees ◽  
Diamond Principle

AbstractWe formulate combinatorial principles that combine the square principle with various strong forms of the diamond principle, and prove that the strongest amongst them holds inLfor every infinite cardinal.As an application, we prove that the following two hold inL:1.For every infinite regular cardinalλ, there exists a special λ+-Aronszajn tree whose projection is almost Souslin;2.For every infinite cardinalλ, there exists arespectingλ+-Kurepa tree; Roughly speaking, this means that this λ+-Kurepa tree looks very much like the λ+-Souslin trees that Jensen constructed inL.

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.

scholarly journals Models of Set Theory in which Nonconstructible Reals First Appear at a Given Projective Level

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 910 ◽  
Author(s):  
Vladimir Kanovei ◽  
Vassily Lyubetsky
Keyword(s):  
Lower Limit ◽  
Set Theory ◽  
Generic Extension ◽  
Projective Class ◽  
Almost Disjoint ◽  
Projective Hierarchy ◽  
Models Of Set Theory

Models of set theory are defined, in which nonconstructible reals first appear on a given level of the projective hierarchy. Our main results are as follows. Suppose that n ≥ 2 . Then: 1. If it holds in the constructible universe L that a ⊆ ω and a ∉ Σ n 1 ∪ Π n 1 , then there is a generic extension of L in which a ∈ Δ n + 1 1 but still a ∉ Σ n 1 ∪ Π n 1 , and moreover, any set x ⊆ ω , x ∈ Σ n 1 , is constructible and Σ n 1 in L . 2. There exists a generic extension L in which it is true that there is a nonconstructible Δ n + 1 1 set a ⊆ ω , but all Σ n 1 sets x ⊆ ω are constructible and even Σ n 1 in L , and in addition, V = L [ a ] in the extension. 3. There exists an generic extension of L in which there is a nonconstructible Σ n + 1 1 set a ⊆ ω , but all Δ n + 1 1 sets x ⊆ ω are constructible and Δ n + 1 1 in L . Thus, nonconstructible reals (here subsets of ω ) can first appear at a given lightface projective class strictly higher than Σ 2 1 , in an appropriate generic extension of L . The lower limit Σ 2 1 is motivated by the Shoenfield absoluteness theorem, which implies that all Σ 2 1 sets a ⊆ ω are constructible. Our methods are based on almost-disjoint forcing. We add a sufficient number of generic reals to L , which are very similar at a given projective level n but discernible at the next level n + 1 .

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].

scholarly journals Closed maximality principles: implications, separations and combinations

2008 ◽  
Vol 73 (1) ◽  
pp. 276-308 ◽  
Author(s):  
Gunter Fuchs
Keyword(s):  
Equivalence Relation ◽  
Regular Cardinal ◽  
Highly Sensitive ◽  
Souslin Trees ◽  
Maximality Principles ◽  
Automorphism Tower

AbstractI investigate versions of the Maximality Principles for the classes of forcings which are <κ-closed, <κ-directed-closed, or of the form Col(κ, <λ). These principles come in many variants, depending on the parameters which are allowed, I shall write MPΓ (A) for the maximality principle for forcings in Γ, with parameters from A. The main results of this paper are:• The principles have many consequences, such as <κ-closed-generic (Hκ) absoluteness, and imply, e.g., that ◊κ holds. I give an application to the automorphism tower problem, showing that there are Souslin trees which are able to realize any equivalence relation, and hence that there are groups whose automorphism tower is highly sensitive to forcing.• The principles can be separated into a hierarchy which is strict, for many κ.• Some of the principles can be combined, in the sense that they can hold at many different κ simultaneously.The possibilities of combining the principles are limited, though: While it is consistent that MP<κ-closed(Hκ +) holds at all regular κ below any fixed α, the “global” maximality principle, stating that MP<κ-closed (Hκ ∪ {κ} ) holds at every regular κ, is inconsistent. In contrast to this, it is equiconsistent with ZFC that the maximality principle for directed-closed forcings without any parameters holds at every regular cardinal. It is also consistent that every local statement with parameters from Hκ⊦ that's provably <κ-closed-forceably necessary is true, for all regular κ.

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.

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