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.

scholarly journals ON SINGULAR STATIONARITY II (TIGHT STATIONARITY AND EXTENDERS-BASED METHODS)

2019 ◽  
Vol 84 (1) ◽  
pp. 320-342
Author(s):  
OMER BEN-NERIA
Keyword(s):  
Generic Extension ◽  
Ground Model ◽  
Regular Cardinals ◽  
Consistency Results ◽  
Image Position ◽  
Model Sequence

AbstractWe study the notion of tightly stationary sets which was introduced by Foreman and Magidor in [8]. We obtain two consistency results showing that certain sequences of regular cardinals ${\langle {\kappa _n}\rangle _{n < \omega }}$ can have the property that in some generic extension, every ground-model sequence of fixed-cofinality stationary sets ${S_n} \subseteq {\kappa _n}$ is tightly stationary. The results are obtained using variations of the short-extenders forcing method.

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.

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 ITERATING SYMMETRIC EXTENSIONS

2019 ◽  
Vol 84 (1) ◽  
pp. 123-159 ◽  
Author(s):  
ASAF KARAGILA
Keyword(s):  
Axiom Of Choice ◽  
Generic Extension ◽  
Ground Model ◽  
Intermediate Model ◽  
Symmetric Extension ◽  
Consistency Results ◽  
New Models ◽  
Image Position ◽  
The Axiom Of Choice

AbstractThe notion of a symmetric extension extends the usual notion of forcing by identifying a particular class of names which forms an intermediate model of $ZF$ between the ground model and the generic extension, and often the axiom of choice fails in these models. Symmetric extensions are generally used to prove choiceless consistency results. We develop a framework for iterating symmetric extensions in order to construct new models of $ZF$. We show how to obtain some well-known and lesser-known results using this framework. Specifically, we discuss Kinna–Wagner principles and obtain some results related to their failure.

Laver indestructibility and the class of compact cardinals

1998 ◽  
Vol 63 (1) ◽  
pp. 149-157 ◽  
Author(s):  
Arthur W. Apter
Keyword(s):  
Limit Point ◽  
Supercompact Cardinal ◽  
Generic Extension ◽  
Ground Model ◽  
Strongly Compact Cardinal ◽  
Joint Work ◽  
Compact Cardinal ◽  
Supercompact Cardinals ◽  
The Universe ◽  
Strongly Compact Cardinals

AbstractUsing an idea developed in joint work with Shelah, we show how to redefine Laver's notion of forcing making a supercompact cardinal κ indestructible under κ-directed closed forcing to give a new proof of the Kimchi-Magidor Theorem in which every compact cardinal in the universe (supercompact or strongly compact) satisfies certain indestructibility properties. Specifically, we show that if K is the class of supercompact cardinals in the ground model, then it is possible to force and construct a generic extension in which the only strongly compact cardinals are the elements of K or their measurable limit points, every κ ∈ K is a supercompact cardinal indestructible under ∈-directed closed forcing, and every κ a measurable limit point of K is a strongly compact cardinal indestructible under κ-directed closed forcing not changing ℘(κ). We then derive as a corollary a model for the existence of a strongly compact cardinal κ which is not κ+ supercompact but which is indestructible under κ-directed closed forcing not changing ℘(κ) and remains non-κ+ supercompact after such a forcing has been done.

The wigner property for CL-spaces and finite-dimensional polyhedral banach spaces

2021 ◽  
pp. 1-17
Author(s):  
Dongni Tan ◽  
Xujian Huang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Phase Function ◽  
Linear Isometry ◽  
Basic Properties ◽  
Finite Dimensional ◽  
Image Position

Abstract We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality \[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \] holds for all $x,\,y\in X$ . A Banach space $X$ is said to have the Wigner property if for any Banach space $Y$ and every surjective phase-isometry $f : X\rightarrow Y$ , there exists a phase function $\varepsilon : X \rightarrow \{-1,\,1\}$ such that $\varepsilon \cdot f$ is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.

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).

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.

scholarly journals A COMPUTABLE FUNCTOR FROM GRAPHS TO FIELDS

2018 ◽  
Vol 83 (1) ◽  
pp. 326-348 ◽  
Author(s):  
RUSSELL MILLER ◽  
BJORN POONEN ◽  
HANS SCHOUTENS ◽  
ALEXANDRA SHLAPENTOKH
Keyword(s):  
Model Theory ◽  
Open Problem ◽  
Category Theory ◽  
Computable Model ◽  
Computable Model Theory ◽  
Arbitrary Characteristic ◽  
Faithful Functor ◽  
New Construction ◽  
Image Position ◽  
Countable Structure

AbstractFried and Kollár constructed a fully faithful functor from the category of graphs to the category of fields. We give a new construction of such a functor and use it to resolve a longstanding open problem in computable model theory, by showing that for every nontrivial countable structure${\cal S}$, there exists a countable field${\cal F}$of arbitrary characteristic with the same essential computable-model-theoretic properties as${\cal S}$. Along the way, we develop a new “computable category theory”, and prove that our functor and its partially defined inverse (restricted to the categories of countable graphs and countable fields) are computable functors.

scholarly journals Are there Ghost Images of the Coma Cluster at other Redshifts?

1999 ◽  
Vol 183 ◽  
pp. 263-263
Author(s):  
Boudewijn F. Roukema
Keyword(s):  
Fundamental Property ◽  
Infinite Volume ◽  
Coma Cluster ◽  
Negatively Curved ◽  
Image Position ◽  
Topology Of The Universe ◽  
The Universe ◽  
Ghost Images

The topology of the Universe is a fundamental property of our Universe according to Friedmann-Lemaître models[3, 9, 7], but has not yet been reliably measured. As pointed out by Sato[12], the Universe may be finite even though flat or negatively curved: infinite volume of a hypersurface.

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