scholarly journals ♣ Does not imply the existence of a Suslin tree

1999 ◽  
Vol 113 (1) ◽  
pp. 163-204 ◽  
Author(s):  
Mirna Džamonja ◽  
Saharon Shelah
Keyword(s):  
Suslin Tree

On Σ1 well-orderings of the universe

1976 ◽  
Vol 41 (1) ◽  
pp. 167-170
Author(s):  
Leo Harrington ◽  
Thomas Jech
Keyword(s):  
Simple Formula ◽  
Generic Extension ◽  
Joint Paper ◽  
Suslin Tree ◽  
The Universe

The constructible universe L of Gödel [2] has a natural well-ordering <L given by the order of construction; a closer look reveals that this well-ordering is definable by a Σ1 formula. Cohen's method of forcing provides several examples of models of ZF + V ≠ L which have a definable well-ordering but none is definable by a relatively simple formula.Recently, Mansfield [7] has shown that if a set of reals (or hereditarily countable sets) has a Σ1, well-ordering then each of its elements is constructible. A question has thus arisen whether one can find a model of ZF + V ≠ L that has a Σ1 well-ordering of the universe. We answer this question in the affirmative.The main result of this paper isTheorem. There is a model of ZF + V ≠ L which has a Σ1 well-ordering.The model is a generic extension of L by adjoining a branch through a Suslin tree with certain properties. The branch is a nonconstructible subset of ℵ1. Note that by Mansfield's theorem, the model must not have nonconstructible subsets of ω.Our results can be generalized in several directions. We note that in particular, we can get a model with a Σ1 well-ordering that is not L[X] for any set X. As one might expect from a joint paper by a recursion theorist and a set theorist, the proof consists of a construction and a computation.

Jensen's ⃞ principles and the Novák number of partially ordered sets

1986 ◽  
Vol 51 (1) ◽  
pp. 47-58 ◽  
Author(s):  
Boban Veličković
Keyword(s):  
Partial Order ◽  
Partially Ordered Sets ◽  
Large Cardinals ◽  
Ordered Sets ◽  
Weakly Compact ◽  
Cardinal Function ◽  
Partially Ordered ◽  
Suslin Tree

In this paper we consider various properties of Jensen's □ principles and use them to construct several examples concerning the so-called Novák number of partially ordered sets.In §1 we give the relevant definitions and review some facts about □ principles. Apart from some simple observations most of the results in this section are known.In §2 we consider the Novák number of partially ordered sets and, using □ principles, give counterexamples to the productivity of this cardinal function. We also formulate a principle, show by forcing that it is consistent and use it to construct an ℵ2-Suslin tree T such that forcing with T × T collapses ℵ1.In §3 we briefly consider games played on partially ordered sets and relate them to the problems of the previous section. Using a version of □ we give an example of a proper partial order such that the game of length ω played on is undetermined.In §4 we raise the question of whether the Novák number of a homogenous partial order can be singular, and show that in some cases the answer is no.We assume familiarity with the basic techniques of forcing. In §1 some facts about large cardinals (e.g. weakly compact cardinals are -indescribable) and elementary properties of the constructible hierarchy are used. For this and all undefined terms we refer the reader to Jech [10].

PRESERVATION OF SUSLIN TREES AND SIDE CONDITIONS

2016 ◽  
Vol 81 (2) ◽  
pp. 483-492
Author(s):  
GIORGIO VENTURI
Keyword(s):  
Countable Support ◽  
Suslin Tree ◽  
Preservation Theorem ◽  
Proper Forcing ◽  
Countable Support Iteration ◽  
Forcing Axiom

AbstractWe show how to force, with finite conditions, the forcing axiom PFA(T), a relativization of PFA to proper forcing notions preserving a given Suslin tree T. The proof uses a Neeman style iteration with generalized side conditions consisting of models of two types, and a preservation theorem for such iterations. The consistency of this axiom was previously known using a standard countable support iteration and a preservation theorem due to Miyamoto.

scholarly journals Forcings constructed along morasses

2011 ◽  
Vol 76 (4) ◽  
pp. 1097-1125 ◽  
Author(s):  
Bernhard Irrgang
Keyword(s):  
Hausdorff Topology ◽  
Simple Observation ◽  
Alternative Construction ◽  
Suslin Tree ◽  
A Chain

AbstractWe further develop a previously introduced method of constructing forcing notions with the help of morasses. There are two new results: (1) If there is a simplified (ω1, 1)-morass, then there exists a ccc forcing of sizeω1that adds an ω2-Suslin tree. (2) If there is a simplified (ω1, 2)-morass, then there exists a ccc forcing of sizeω1that adds a 0-dimensional Hausdorff topologyτonω3which has spreads(τ) =ω1. While (2) is the main result of the paper, (1) is only an improvement of a previous result, which is based on a simple observation. Both forcings preserveGCH. To show that the method can be changed to produce models where CH fails, we give an alternative construction of Koszmider's model in which there is a chain 〈Xα∣α<ω2〉 such thatXα⊆ω1.Xβ–Xαis finite andXα–Xβhas sizeω1for allβ<α<ω2.

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.

scholarly journals Rudin's Dowker space in the extension with a Suslin tree

2008 ◽  
Vol 201 (1) ◽  
pp. 53-89 ◽  
Author(s):  
Teruyuki Yorioka
Keyword(s):  
Suslin Tree ◽  
Dowker Space

Embedding theorems for boolean algebras and consistency results on ordinal definable sets

1977 ◽  
Vol 42 (1) ◽  
pp. 64-76 ◽  
Author(s):  
Petr Štěpánek ◽  
Bohuslav Balcar
Keyword(s):  
Boolean Algebra ◽  
Extreme Case ◽  
Chain Condition ◽  
Boolean Algebras ◽  
Complete Boolean Algebra ◽  
Suslin Tree ◽  
Consistency Results ◽  
Weak Homogeneity ◽  
Uncountable Cardinal ◽  
Definable Sets

The existence of complete rigid Boolean algebras was first proved by McAloon [8] who also showed that every Boolean algebra can be completely embedded in a rigid complete Boolean algebra. McAloon was interested in consistency results on ordinal definable sets. His approach was based on forcing. Recently, Shelah [10] proved that for every uncountable cardinal κ there exists a Boolean algebra of power κ with rigid completion. Extending his method, we get the following theorems.Theorem 1. Any Boolean algebra B can be completely embedded in a complete Boolean algebra C with no nontrivial σ-complete one-one endomor-phism. If B satisfies the κ-chain condition for an uncountable cardinal κ, the same holds true for C.Since every automorphism is a complete endomorphism, it follows from Theorem 1 that C is rigid. The other extreme case of Boolean algebras are homogeneous algebras. It was proved by Kripke [7] that every Boolean algebra can be completely embedded in a homogeneous complete Boolean algebra. In his proof, the homogeneous algebra contains antichains of cardinality equal to the power of the embedded Boolean algebra. The following result shows that this is essential: the analogue of Theorem 1 is not provable in set theory even for Boolean algebras with a very weak homogeneity property. We use a Suslin tree with particular properties constructed by Jensen [6] in conjunction with a forcing argument.

A Corson compact L-space from a Suslin tree

2015 ◽  
Vol 141 (2) ◽  
pp. 149-155
Author(s):  
Peter Nyikos
Keyword(s):  
Suslin Tree ◽  
Corson Compact
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