Changing cofinalities and infinite exponents

1981 ◽  
Vol 46 (1) ◽  
pp. 89-95 ◽  
Author(s):  
Arthur W. Apter
Keyword(s):  
Early Result ◽  
Regular Cardinal ◽  
Generic Extension ◽  
Ground Model ◽  
Generic Extensions

Ever since Cohen invented forcing in 1963, people have studied the properties that cardinals can have in generic extensions of the ground model. A very early result of Lévy shows that if κ is a regular cardinal and λ > κ is strongly inaccessible, then there is a notion of forcing which collapses every cardinal strictly between κ and λ yet preserves every other cardinal. This, of course, answers one question of the genre “What properties can a cardinal have in a generic extension?”Another question of the same genre that can be asked is the following: Is it possible to have a generic extension of the ground model in which all cardinals are preserved and yet the cofinalities of some cardinals are different? This question was first answered in the affirmative by Prikry, who proved the following theorem.Theorem 1.1 (Prikry [5]). Assume that V ⊨ “ZFC + κ is measurable”. Then there is a notion of forcing, P, such that for G V-generic over P:(1) V and V[G] have the same cardinals.(2) V and V[G] have the same bounded subsets of κ.(3) V[G], i.e, V[G] ⊨ “κ is Rowbottom”.(4) V[G] ⊨ “cof(κ) = ω”.Prikry's result naturally raises the following question: Is it possible to get a generic extension in which cardinals are preserved and yet the cofinalities of certain cardinals are different from the ground model's but are uncountable? This question was first answered in the affirmative by Magidor, who proved the following theorem.

scholarly journals COMPUTABLE STRUCTURES IN GENERIC EXTENSIONS

2016 ◽  
Vol 81 (3) ◽  
pp. 814-832 ◽  
Author(s):  
JULIA KNIGHT ◽  
ANTONIO MONTALBÁN ◽  
NOAH SCHWEBER
Keyword(s):  
Positive Result ◽  
Generic Extension ◽  
Ground Model ◽  
Rigid Structure ◽  
Basic Properties ◽  
Image Position ◽  
Scott Rank ◽  
The Universe ◽  
Generic Extensions ◽  
Countable Structure

AbstractIn this paper, we investigate connections between structures present in every generic extension of the universe V and computability theory. We introduce the notion of generic Muchnik reducibility that can be used to compare the complexity of uncountable structures; we establish basic properties of this reducibility, and study it in the context of generic presentability, the existence of a copy of the structure in every extension by a given forcing. We show that every forcing notion making ω2 countable generically presents some countable structure with no copy in the ground model; and that every structure generically presentable by a forcing notion that does not make ω2 countable has a copy in the ground model. We also show that any countable structure ${\cal A}$ that is generically presentable by a forcing notion not collapsing ω1 has a countable copy in V, as does any structure ${\cal B}$ generically Muchnik reducible to a structure ${\cal A}$ of cardinality ℵ1. The former positive result yields a new proof of Harrington’s result that counterexamples to Vaught’s conjecture have models of power ℵ1 with Scott rank arbitrarily high below ω2. Finally, we show that a rigid structure with copies in all generic extensions by a given forcing has a copy already in the ground model.

-definability at uncountable regular cardinals

2012 ◽  
Vol 77 (3) ◽  
pp. 1011-1046 ◽  
Author(s):  
Philipp Lücke
Keyword(s):  
Partial Order ◽  
Regular Cardinal ◽  
Short Proof ◽  
Chain Condition ◽  
Generic Extension ◽  
Infinite Cardinal ◽  
Ground Model ◽  
Generic Absoluteness ◽  
Regular Cardinals ◽  
Quasi Order

AbstractLet κ be an infinite cardinal. A subset of (κκ)n is a -subset if it is the projection p[T] of all cofinal branches through a subtree T of (>κκ)n+1 of height κ. We define and -subsets of (κκ)n as usual.Given an uncountable regular cardinal κ with κ = κ<κ and an arbitrary subset A of κκ, we show that there is a <κ-closed forcing ℙ that satisfies the κ+-chain condition and forces A to be a -subset of κκ in every ℙ-generic extension of V. We give some applications of this result and the methods used in its proof.(i) Given any set x, we produce a partial order with the above properties that forces x to be an element of L.(ii) We show that there is a partial order with the above properties forcing the existence of a well-ordering of κκ whose graph is a -subset of κκ × κκ.(iii) We provide a short proof of a result due to Mekler and Väänänen by using the above forcing to add a tree T of cardinality and height κ such that T has no cofinal branches and every tree from the ground model of cardinality and height κ without a cofinal branch quasi-order embeds into T.(iv) We will show that generic absoluteness for -formulae (i.e., formulae with parameters which define -subsets of κκ) under <κ-closed forcings that satisfy the κ+-chain condition is inconsistent.In another direction, we use methods from the proofs of the above results to show that - and -subsets have some useful structural properties in certain ZFC-models.

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.

Generic Σ31 absoluteness

2004 ◽  
Vol 69 (1) ◽  
pp. 73-80 ◽  
Author(s):  
Sy D. Friedman
Keyword(s):  
Regular Cardinal ◽  
Forcing Axioms ◽  
Proper Forcing ◽  
Class Forcing ◽  
The Universe ◽  
Generic Extensions

In this article we study the strength of absoluteness (with real parameters) in various types of generic extensions, correcting and improving some results from [3]. (In particular, see Theorem 3 below.) We shall also make some comments relating this work to the bounded forcing axioms BMM, BPFA and BSPFA.The statement “ absoluteness holds for ccc forcing” means that if a formula with real parameters has a solution in a ccc set-forcing extension of the universe V, then it already has a solution in V. The analogous definition applies when ccc is replaced by other set-forcing notions, or by class-forcing.Theorem 1. [1] absoluteness for ccc has no strength; i.e., if ZFC is consistent then so is ZFC + absoluteness for ccc.The following results concerning (arbitrary) set-forcing and class-forcing can be found in [3].Theorem 2 (Feng-Magidor-Woodin). (a) absoluteness for arbitrary set-forcing is equiconsistent with the existence of a reflecting cardinal, i.e., a regular cardinal κ such that H(κ) is ∑2-elementary in V.(b) absoluteness for class-forcing is inconsistent.We consider next the following set-forcing notions, which lie strictly between ccc and arbitrary set-forcing: proper, semiproper, stationary-preserving and ω1-preserving. We refer the reader to [8] for the definitions of these forcing notions.Using a variant of an argument due to Goldstern-Shelah (see [6]), we show the following. This result corrects Theorem 2 of [3] (whose proof only shows that if absoluteness holds in a certain proper forcing extension, then in L either ω1 is Mahlo or ω2 is inaccessible).

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 A Reduction of the NF consistency Problem

2007 ◽  
Vol 72 (1) ◽  
pp. 285-304 ◽  
Author(s):  
Athanassios Tzouvaras
Keyword(s):  
Generic Extension ◽  
Sufficient Condition ◽  
Ground Model ◽  
Necessary And Sufficient Condition ◽  
Coherent Pair ◽  
Consistency Problem ◽  
Coherent Pairs ◽  
Necessary And Sufficient ◽  
Type Shifting

AbstractWe give a necessary and sufficient condition in order that a type-shifting automorphism be constructed on a model of the Theory of Simple Types (TST) by forcing. Namely it is proved that, if for every n ≥ 1 there is a model of TST in the ground model M of ZFC that contains an n-extendible coherent pair, then there is a generic extension M[G] of M that contains a model of TST with a type-shifting automorphism, and hence M[G] contains a model of NF. The converse holds trivially. It is also proved that there exist models of TST containing 1-extendible coherent pairs.

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.

Hechler's Theorem for tall analytic P-ideals

2011 ◽  
Vol 76 (2) ◽  
pp. 729-736
Author(s):  
Barnabás Farkas
Keyword(s):  
Upper Bound ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Countable Subset ◽  
Generic Extension ◽  
Classical Theorem ◽  
Ground Model ◽  
Partially Ordered

AbstractWe prove the following version of Hechler's classical theorem: For each partially ordered set (Q, ≤) with the property that every countable subset of Q has a strict upper bound in Q, there is a ccc forcing notion such that in the generic extension for each tall analytic P-ideal (coded in the ground model) a cofinal subset of is order isomorphic to (Q, ≤).

scholarly journals The special Aronszajn tree property

2019 ◽  
Vol 20 (01) ◽  
pp. 2050003 ◽  
Author(s):  
Mohammad Golshani ◽  
Yair Hayut
Keyword(s):  
Regular Cardinal ◽  
Generic Extension ◽  
Proper Class ◽  
Tree Property ◽  
Supercompact Cardinals ◽  
Aronszajn Tree ◽  
Aronszajn Trees

Assuming the existence of a proper class of supercompact cardinals, we force a generic extension in which, for every regular cardinal [Formula: see text], there are [Formula: see text]-Aronszajn trees, and all such trees are special.

scholarly journals LOCAL CLUB CONDENSATION AND L-LIKENESS

2015 ◽  
Vol 80 (4) ◽  
pp. 1361-1378
Author(s):  
PETER HOLY ◽  
PHILIP WELCH ◽  
LIUZHEN WU
Keyword(s):  
Regular Cardinal ◽  
Large Cardinal ◽  
Ground Model ◽  
The Third ◽  
Chang’S Conjecture ◽  
Chang's Conjecture ◽  
Generalized Condensation

AbstractWe present a forcing to obtain a localized version of Local Club Condensation, a generalized Condensation principle introduced by Sy Friedman and the first author in [3] and [5]. This forcing will have properties nicer than the forcings to obtain this localized version that could be derived from the forcings presented in either [3] or [5]. We also strongly simplify the related proofs provided in [3] and [5]. Moreover our forcing will be capable of introducing this localized principle at κ while simultaneously performing collapses to make κ become the successor of any given smaller regular cardinal. This will be particularly useful when κ has large cardinal properties in the ground model. We will apply this to measure how much L-likeness is implied by Local Club Condensation and related principles. We show that Local Club Condensation at κ+ is consistent with ¬☐κ whenever κ is regular and uncountable, generalizing and improving a result of the third author in [14], and that if κ ≥ ω2 is regular, CC(κ+) - Chang’s Conjecture at κ+ - is consistent with Local Club Condensation at κ+, both under suitable large cardinal consistency assumptions.

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