Generic extensions which do not ADD random reals

Generic Extensions of Prinjective Modules

Algebras and Representation Theory ◽

10.1007/s10468-006-9033-2 ◽

2006 ◽

Vol 9 (6) ◽

pp. 557-568 ◽

Cited By ~ 2

Author(s):

Justyna Kosakowska

Keyword(s):

Generic Extensions

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Generic Σ31 absoluteness

Journal of Symbolic Logic ◽

10.2178/jsl/1080938826 ◽

2004 ◽

Vol 69 (1) ◽

pp. 73-80 ◽

Cited By ~ 1

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).

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Saturation properties of ideals in generic extensions. II

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-1982-0654852-4 ◽

1982 ◽

Vol 271 (2) ◽

pp. 587-587

Author(s):

James E. Baumgartner ◽

Alan D. Taylor

Keyword(s):

Saturation Properties ◽

Generic Extensions

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Generic Extensions and Canonical Bases for Cyclic Quivers

Canadian Journal of Mathematics ◽

10.4153/cjm-2007-054-7 ◽

2007 ◽

Vol 59 (6) ◽

pp. 1260-1283 ◽

Cited By ~ 15

Author(s):

Bangming Deng ◽

Jie Du ◽

Jie Xiao

Keyword(s):

Hall Algebra ◽

Quiver Representations ◽

Algebraic Construction ◽

Positive Part ◽

Monomial Basis ◽

Canonical Bases ◽

Generic Extensions ◽

Cyclic Quiver

AbstractWe use the monomial basis theory developed by Deng and Du to present an elementary algebraic construction of the canonical bases for both the Ringel–Hall algebra of a cyclic quiver and the positive part U+of the quantum affine. This construction relies on analysis of quiver representations and the introduction of a new integral PBW–like basis for the Lusztig ℤ[v,v–1]-form of U+.

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CHAITIN’S Ω AS A CONTINUOUS FUNCTION

Journal of Symbolic Logic ◽

10.1017/jsl.2019.60 ◽

2019 ◽

Vol 85 (1) ◽

pp. 486-510

Author(s):

RUPERT HÖLZL ◽

WOLFGANG MERKLE ◽

JOSEPH MILLER ◽

FRANK STEPHAN ◽

LIANG YU

Keyword(s):

Continuous Function ◽

Hausdorff Dimension ◽

Universal Machine ◽

Image Position ◽

Turing Complete ◽

Random Reals

AbstractWe prove that the continuous function${\rm{\hat \Omega }}:2^\omega \to $ that is defined via$X \mapsto \mathop \sum \limits_n 2^{ - K\left( {Xn} \right)} $ for all $X \in {2^\omega }$ is differentiable exactly at the Martin-Löf random reals with the derivative having value 0; that it is nowhere monotonic; and that $\mathop \smallint \nolimits _0^1{\rm{\hat{\Omega }}}\left( X \right)\,{\rm{d}}X$ is a left-c.e. $wtt$-complete real having effective Hausdorff dimension ${1 / 2}$.We further investigate the algorithmic properties of ${\rm{\hat{\Omega }}}$. For example, we show that the maximal value of ${\rm{\hat{\Omega }}}$ must be random, the minimal value must be Turing complete, and that ${\rm{\hat{\Omega }}}\left( X \right) \oplus X{ \ge _T}\emptyset \prime$ for every X. We also obtain some machine-dependent results, including that for every $\varepsilon > 0$, there is a universal machine V such that ${{\rm{\hat{\Omega }}}_V}$ maps every real X having effective Hausdorff dimension greater than ε to a real of effective Hausdorff dimension 0 with the property that $X{ \le _{tt}}{{\rm{\hat{\Omega }}}_V}\left( X \right)$; and that there is a real X and a universal machine V such that ${{\rm{\Omega }}_V}\left( X \right)$ is rational.

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Distributive and related ideals in generic extensions

Nagoya Mathematical Journal ◽

10.1017/s0027763000000891 ◽

1987 ◽

Vol 106 ◽

pp. 91-100

Author(s):

C. A. Johnson

Keyword(s):

Quotient Algebra ◽

Complete Ideal ◽

Naturally Occurring ◽

Uncountable Cardinal ◽

Game Theoretic ◽

Generic Extensions

Let κ: be a regular uncountable cardinal and I a κ-complete ideal on te. In [11] Kanai proved that the μ-distributivity of the quotient algebra P(κ)I is preserved under κ-C.C. μ-closed forcing. In this paper we extend Kanai’s result and also prove similar preservation results for other naturally occurring forms of distributivity. We also consider the preservation of two game theoretic properties of I and in particular, using a game theoretic equivalent of precipitousness we give a new proof of Kakuda’s theorem ([10]) that the precipitousness of I is preserved under κ-C.C. forcing.

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From index sets to randomness in ∅n: random reals and possibly infinite computations part II

Journal of Symbolic Logic ◽

10.2178/jsl/1231082305 ◽

2009 ◽

Vol 74 (1) ◽

pp. 124-156 ◽

Cited By ~ 7

Author(s):

Verónica Becher ◽

Serge Grigorieff

Keyword(s):

Large Class ◽

Turing Machine ◽

Index Sets ◽

Random Reals

AbstractWe obtain a large class of significant examples of n-random reals (i.e., Martin-Löf random in oracle ∅(n−1)) à la Chaitin. Any such real is defined as the probability that a universal monotone Turing machine performing possibly infinite computations on infinite (resp. finite large enough, resp. finite self-delimited) inputs produces an output in a given set . In particular, we develop methods to transfer many-one completeness results of index sets to n-randomness of associated probabilities.

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