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16
Properness without elementaricity
, 2003
"... We present reasons for developing a theory of forcing notions which satisfy the properness demand for countable models which are not necessarily elementary submodels of some (H(χ), ∈). This leads to forcing notions which are “reasonably ” definable. We present two specific properties materializing ..."
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We present reasons for developing a theory of forcing notions which satisfy the properness demand for countable models which are not necessarily elementary submodels of some (H(χ), ∈). This leads to forcing notions which are “reasonably ” definable. We present two specific properties materializing this intuition: nep (nonelementary properness) and snep (Souslin nonelementary properness) and also the older Souslin proper. For this we consider candidates (countable models to which the definition applies). A major theme here is “preservation by iteration”, but we also show a dichotomy: if such forcing notions preserve the positiveness of the set of old reals for some naturally defined c.c.c. ideal, then they preserve the positiveness of any old positive set hence preservation by composition of two follows. Last but not least, we prove that (among such forcing notions) the only one commuting with Cohen is Cohen itself; in other words, any other such forcing notion make the set of old reals to a meager set. In the end we present some open problems in this area.
Between Maharam’s and von Neumann’s problems
, 2004
"... In the context of definable algebras Maharam’s and von Neumann’s problems essentially coincide. Consequently, random forcing is the only definable ccc forcing adding a single real that does not make the ground model reals null, and the only pairs of definable ccc σideals with the Fubini property ar ..."
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In the context of definable algebras Maharam’s and von Neumann’s problems essentially coincide. Consequently, random forcing is the only definable ccc forcing adding a single real that does not make the ground model reals null, and the only pairs of definable ccc σideals with the Fubini property are (meager, meager) and (null, null). In Scottish Book, von Neumann asked whether every ccc, weakly distributive complete Boolean algebra carries a strictly positive probability measure. Von Neumann’s problem naturally splits into two: (a) whether all such algebras carry a strictly positive continuous submeasure, and (b) whether every algebra that carries a strictly positive continuous submeasure carries a strictly positive measure. The latter problem is known under the names of Maharam’s Problem and Control Measure Problem (see [16], [9], [5, §393]). While von Neumann’s problem has a consistently negative answer ([16]), Maharam’s problem can be stated as a Σ12 statement and is therefore, by Shoenfield’s theorem, absolute between transitive models of set theory containing all countable ordinals. Theorem 0.1. Let I be a c.c.c. σideal on Borel subsets of 2ω that is analytic on Gδ. The following are equivalent: • PI is a weakly distributive notion of forcing • there is a continuous submeasure on 2ω such that I is the σideal of its null sets. A suitable large cardinal assumption implies that the assumption ‘I is analytic on Gδ ’ can be relaxed to ‘I is definable.’
On nicely definable forcing notions
 Journal of Applied Analysis
"... Abstract. We prove that if Q is a nwnep forcing then it cannot add a dominating real. We also show that amoeba forcing cannot be P(X)/I if I is an ℵ1complete ideal. Furthermore, we generalize the results of [12]. Nicely definable forcing notions have been studied since the mideighties, especially ..."
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Abstract. We prove that if Q is a nwnep forcing then it cannot add a dominating real. We also show that amoeba forcing cannot be P(X)/I if I is an ℵ1complete ideal. Furthermore, we generalize the results of [12]. Nicely definable forcing notions have been studied since the mideighties, especially for the case when “nicely definable ” was interpreted as “Souslin” (see, e.g., [12], Judah and Shelah [8] or Goldstern and Judah [7]). Recently, in [14], we have initialized investigations of a wide class of “reasonably”
ASYMPTOTIC UPPER BOUNDS ON THE SHADES OF tINTERSECTING FAMILIES
, 2008
"... We examine the mshades of tintersecting families of ksubsets of [n], and conjecture on the optimal upper bound on their cardinalities. This conjecture extends Frankl’s General Conjecture that was proven true by Ahlswede–Khachatrian. From this we deduce the precise asymptotic upper bounds on the ..."
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We examine the mshades of tintersecting families of ksubsets of [n], and conjecture on the optimal upper bound on their cardinalities. This conjecture extends Frankl’s General Conjecture that was proven true by Ahlswede–Khachatrian. From this we deduce the precise asymptotic upper bounds on the cardinalities of mshades of t(m)intersecting families of k(m)subsets of [2m], as m → ∞. A generalization to crosstintersecting families is also considered.
MAHARAM ALGEBRAS AND COHEN REALS
, 2007
"... We show that the product of any two nonatomic Maharam algebras adds a Cohen real. As a corollary of this and a result of Shelah (1994) we obtain that the product of any two nonatomic ccc Souslin forcing notions adds a Cohen real. ..."
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We show that the product of any two nonatomic Maharam algebras adds a Cohen real. As a corollary of this and a result of Shelah (1994) we obtain that the product of any two nonatomic ccc Souslin forcing notions adds a Cohen real.
Was Sierpiński right? III Can continuum–c.c. times c.c.c. be continuum–c.c
 Annals of Pure and Applied Logic, 78:259–269
, 1996
"... Abstract. We prove the consistency of: if B1, B2 are Boolean algebras satisfying the c.c.c. and the 2 ℵ0c.c. respectively then B1 × B2 satisfies the 2 ℵ0c.c. We start with a universe with a Ramsey cardinal (less suffice). (481) revision:19950915 modified:19990904 Typed 5/92 Latest Revision 1/ ..."
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Abstract. We prove the consistency of: if B1, B2 are Boolean algebras satisfying the c.c.c. and the 2 ℵ0c.c. respectively then B1 × B2 satisfies the 2 ℵ0c.c. We start with a universe with a Ramsey cardinal (less suffice). (481) revision:19950915 modified:19990904 Typed 5/92 Latest Revision 1/13/95 I thank Alice Leonhardt for the beautiful typing §1 corrected 4/94 Partially supported by the basic research fund, Israeli Academy 1 Typeset by AMSTEX2 SAHARON SHELAH
The hyperweak distributive law and a related game in Boolean algebras
, 2007
"... We discuss the relationship between various weak distributive laws and games in Boolean algebras. In the first part we give some game characterizations for certain forms of Prikry’s “hyperweak distributive laws”, and in the second part we construct Suslin algebras in which neither player wins a cer ..."
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We discuss the relationship between various weak distributive laws and games in Boolean algebras. In the first part we give some game characterizations for certain forms of Prikry’s “hyperweak distributive laws”, and in the second part we construct Suslin algebras in which neither player wins a certain hyperweak distributivity game. We conclude that in the constructible universe L, all the distributivity games considered in this paper may be undetermined.
Maharam Algebras
, 2008
"... Maharam algebras are complete Boolean algebras carrying a positive continuous submeasure. They were introduced and studied by Maharam in [24] in relation to Von Neumann’s problem on the characterization of measure algebras. The question whether every Maharam algebra is a measure algebra has been the ..."
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Maharam algebras are complete Boolean algebras carrying a positive continuous submeasure. They were introduced and studied by Maharam in [24] in relation to Von Neumann’s problem on the characterization of measure algebras. The question whether every Maharam algebra is a measure algebra has been the main open problem in this area for around 60 years. It was finally resolved by Talagrand [31] who provided the first example of a Maharam algebra which is not a measure algebra. In this paper we survey some recent work on Maharam algebras in relation to the two conditions proposed by Von Neumann: weak distributivity and the countable chain condition. It turns out that by strengthening either one of these conditions one obtains a ZFC characterization of Maharam algebras. We also present some results on Maharam algebras as forcing notions showing that they share some of the well known properties of measure algebras.