HIERARCHIES OF FORCING AXIOMS, THE CONTINUUM HYPOTHESIS AND SQUARE PRINCIPLES

2018 ◽  
Vol 83 (1) ◽  
pp. 256-282 ◽  
Author(s):  
GUNTER FUCHS
Keyword(s):  
Continuum Hypothesis ◽  
Reflection Principle ◽  
Forcing Axioms ◽  
Subcomplete Forcing ◽  
Square Principles ◽  
Consistency Strengths ◽  
Remarkable Cardinals ◽  
Forcing Axiom ◽  
The Continuum

AbstractI analyze the hierarchies of the bounded and the weak bounded forcing axioms, with a focus on their versions for the class of subcomplete forcings, in terms of implications and consistency strengths. For the weak hierarchy, I provide level-by-level equiconsistencies with an appropriate hierarchy of partially remarkable cardinals. I also show that the subcomplete forcing axiom implies Larson’s ordinal reflection principle atω2, and that its effect on the failure of weak squares is very similar to that of Martin’s Maximum.

scholarly journals Canonical fragments of the strong reflection principle

2021 ◽  
pp. 2150023
Author(s):  
Gunter Fuchs
Keyword(s):  
Reflection Principle ◽  
Forcing Axioms ◽  
Subcomplete Forcing ◽  
Text Fragment ◽  
Strong Reflection ◽  
Stationary Set ◽  
Mutual Stationarity ◽  
Forcing Axiom ◽  
The Way

For an arbitrary forcing class [Formula: see text], the [Formula: see text]-fragment of Todorčević’s strong reflection principle SRP is isolated in such a way that (1) the forcing axiom for [Formula: see text] implies the [Formula: see text]-fragment of SRP , (2) the stationary set preserving fragment of SRP is the full principle SRP , and (3) the subcomplete fragment of SRP implies the major consequences of the subcomplete forcing axiom. This fragment of SRP is consistent with CH , and even with Jensen’s principle [Formula: see text]. Along the way, some hitherto unknown effects of (the subcomplete fragment of) SRP on mutual stationarity are explored, and some limitations to the extent to which fragments of SRP may capture the effects of their corresponding forcing axioms are established.

scholarly journals Rado’s Conjecture and its Baire version

2019 ◽  
Vol 20 (01) ◽  
pp. 1950015
Author(s):  
Jing Zhang
Keyword(s):  
Reflection Principle ◽  
Stationary Reflection ◽  
Forcing Axioms ◽  
Martin’S Axiom ◽  
Partition Relations ◽  
Consistency Results ◽  
Square Principles ◽  
Reflection Principles ◽  
Weak Square ◽  
Forcing Axiom

Rado’s Conjecture is a compactness/reflection principle that says any nonspecial tree of height [Formula: see text] has a nonspecial subtree of size [Formula: see text]. Though incompatible with Martin’s Axiom, Rado’s Conjecture turns out to have many interesting consequences that are also implied by certain forcing axioms. In this paper, we obtain consistency results concerning Rado’s Conjecture and its Baire version. In particular, we show that a fragment of [Formula: see text], which is the forcing axiom for Baire Indestructibly Proper forcings, is compatible with the Baire Rado’s Conjecture. As a corollary, the Baire Rado’s Conjecture does not imply Rado’s Conjecture. Then we discuss the strength and limitations of the Baire Rado’s Conjecture regarding its interaction with stationary reflection principles and some families of weak square principles. Finally, we investigate the influence of Rado’s Conjecture on some polarized partition relations.

Forcing for the impredicative theory of classes

1972 ◽  
Vol 37 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Rolando Chuaqui
Keyword(s):  
General Theory ◽  
Set Theory ◽  
Continuum Hypothesis ◽  
Reflection Principle ◽  
Inaccessible Cardinal ◽  
Consistency Proof ◽  
Suitable Form ◽  
Definition Of ◽  
Class Construction ◽  
The Continuum

The purpose of this work is to formulate a general theory of forcing with classes and to solve some of the consistency and independence problems for the impredicative theory of classes, that is, the set theory that uses the full schema of class construction, including formulas with quantification over proper classes. This theory is in principle due to A. Morse [9]. The version I am using is based on axioms by A. Tarski and is essentially the same as that presented in [6, pp. 250–281] and [10, pp. 2–11]. For a detailed exposition the reader is referred there. This theory will be referred to as .The reflection principle (see [8]), valid for other forms of set theory, is not provable in . Some form of the reflection principle is essential for the proofs in the original version of forcing introduced by Cohen [2] and the version introduced by Mostowski [10]. The same seems to be true for the Boolean valued models methods due to Scott and Solovay [12]. The only suitable form of forcing for found in the literature is the version that appears in Shoenfield [14]. I believe Vopěnka's methods [15] would also be applicable. The definition of forcing given in the present paper is basically derived from Shoenfield's definition. Shoenfield, however, worked in Zermelo-Fraenkel set theory.I do not know of any proof of the consistency of the continuum hypothesis with assuming only that is consistent. However, if one assumes the existence of an inaccessible cardinal, it is easy to extend Gödel's consistency proof [4] of the axiom of constructibility to .

Continuum-many Boolean algebras of the form Borel

2004 ◽  
Vol 69 (3) ◽  
pp. 799-816 ◽  
Author(s):  
Michael Ray Oliver
Keyword(s):  
Equivalence Relation ◽  
Continuum Hypothesis ◽  
Boolean Algebras ◽  
New Technique ◽  
Finite Sets ◽  
A New Technique ◽  
Borel Ideals ◽  
Forcing Axiom ◽  
The Continuum ◽  
The Ideal

Abstract.We examine the question of how many Boolean algebras, distinct up to isomorphism, that are quotients of the powerset of the naturals by Borel ideals, can be proved to exist in ZFC alone. The maximum possible value is easily seen to be the cardinality of the continuum ; earlier work by Ilijas Farah had shown that this was the value in models of Martin's Maximum or some similar forcing axiom, but it was open whether there could be fewer in models of the Continuum Hypothesis.We develop and apply a new technique for constructing many ideals whose quotients must be nonisomorphic in any model of ZFC. The technique depends on isolating a kind of ideal, called shallow, that can be distinguished from the ideal of all finite sets even after any isomorphic embedding, and then piecing together various copies of the ideal of all finite sets using distinct shallow ideals. In this way we are able to demonstrate that there are continuum-many distinct quotients by Borel ideals, indeed by analytic P-ideals, and in fact that there is in an appropriate sense a Borel embedding of the Vitali equivalence relation into the equivalence relation of isomorphism of quotients by analytic P-ideals. We also show that there is an uncountable definable wellordered collection of Borel ideals with distinct quotients.

scholarly journals Forcing axioms and the continuum hypothesis

2013 ◽  
Vol 210 (1) ◽  
pp. 1-29 ◽  
Author(s):  
David Asperό ◽  
Paul Larson ◽  
Justin Tatch Moore
Keyword(s):  
Continuum Hypothesis ◽  
Forcing Axioms ◽  
The Continuum

Completely Additive Liftings

1998 ◽  
Vol 4 (1) ◽  
pp. 37-54 ◽  
Author(s):  
Ilijas Farah
Keyword(s):  
Set Theory ◽  
Classical Result ◽  
Continuum Hypothesis ◽  
Axiomatic Approach ◽  
Proper Forcing Axiom ◽  
And Topology ◽  
Proper Forcing ◽  
Forcing Axiom ◽  
The Continuum ◽  
The Ideal

The purpose of this communication is to survey a theory of liftings, as developed in author's thesis ([8]). The first result in this area was Shelah's construction of a model of set theory in which every automorphism of P(ℕ)/ Fin, where Fin is the ideal of finite sets, is trivial, or inother words, it is induced by a function mapping integers into integers ([33]). (It is a classical result of W. Rudin [31] that under the Continuum Hypothesis there are automorphisms other than trivial ones.) Soon afterwards, Velickovic ([47]), was able to extract from Shelah's argument the fact that every automorphism of P(ℕ)/ Fin with a Baire-measurable lifting has to be trivial. This, for instance, implies that in Solovay's model ([36]) all automorphisms are trivial. Later on, an axiomatic approach was adopted and Shelah's conclusion was drawn first from the Proper Forcing Axiom (PFA) ([34]) and then from the milder Open Coloring Axiom (OCA) and Martin's Axiom (MA) ([48], see §5 for definitions). Both shifts from the quotient P(ℕ)/ Fin to quotients over more general ideals P(ℕ)/I and from automorphisms to arbitrary ho-momorphisms were made by Just in a series of papers ([14]-[17]), motivated by some problems in algebra ([7, pp. 38–39], [43, I.12.11], [45, Q48]) and topology ([46, p. 537]).

scholarly journals SUBCOMPLETE FORCING, TREES, AND GENERIC ABSOLUTENESS

2018 ◽  
Vol 83 (3) ◽  
pp. 1282-1305 ◽  
Author(s):  
GUNTER FUCHS ◽  
KAETHE MINDEN
Keyword(s):  
Forcing Axioms ◽  
First Order ◽  
Generic Absoluteness ◽  
Subcomplete Forcing ◽  
Image Position ◽  
Aronszajn Trees ◽  
Order Structures ◽  
Forcing Axiom

AbstractWe investigate properties of trees of height ω1 and their preservation under subcomplete forcing. We show that subcomplete forcing cannot add a new branch to an ω1-tree. We introduce fragments of subcompleteness which are preserved by subcomplete forcing, and use these in order to show that certain strong forms of rigidity of Suslin trees are preserved by subcomplete forcing. Finally, we explore under what circumstances subcomplete forcing preserves Aronszajn trees of height and width ω1. We show that this is the case if CH fails, and if CH holds, then this is the case iff the bounded subcomplete forcing axiom holds. Finally, we explore the relationships between bounded forcing axioms, preservation of Aronszajn trees of height and width ω1 and generic absoluteness of ${\rm{\Sigma }}_1^1$-statements over first order structures of size ω1, also for other canonical classes of forcing.

Forcing

2018 ◽  
Author(s):  
John P. Burgess
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Axiom Of Choice ◽  
Forcing Axioms ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
The Axiom Of Choice ◽  
The Continuum

The method of forcing was introduced by Paul J. Cohen in order to prove the independence of the axiom of choice (AC) from the basic (ZF) axioms of set theory, and of the continuum hypothesis (CH) from the accepted axioms (ZFC = ZF + AC) of set theory (see set theory, axiom of choice, continuum hypothesis). Given a model M of ZF and a certain P∈M, it produces a ‘generic’ G⊆P and a model N of ZF with M⊆N and G∈N. By suitably choosing P, N can be ‘forced’ to be or not be a model of various hypotheses, which are thus shown to be consistent with or independent of the axioms. This method of proving undecidability has been very widely applied. The method has also motivated the proposal of new so-called forcing axioms to decide what is otherwise undecidable, the most important being that called Martin’s axiom (MA).

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