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16
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”
MEASURED CREATURES
"... Abstract. We prove that two basic questions on outer measure are undecidable. First we show that consistently • every supmeasurable function f: R 2 − → R is measurable. The interest in supmeasurable functions comes from differential equations and the question for which functions f: R 2 − → R the C ..."
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Abstract. We prove that two basic questions on outer measure are undecidable. First we show that consistently • every supmeasurable function f: R 2 − → R is measurable. The interest in supmeasurable functions comes from differential equations and the question for which functions f: R 2 − → R the Cauchy problem y ′ = f(x, y), y(x0) = y0 has a unique almosteverywhere solution in the class ACl(R) of locally absolutely continuous functions on R. Next we prove that consistently • every function f: R − → R is continuous on some set of positive outer Lebesgue measure. This says that in a strong sense the family of continuous functions (from the reals to the reals) is dense in the space of arbitrary such functions. For the proofs we discover and investigate a new family of nicely definable
LARGE CONTINUUM, ORACLES
, 2009
"... Our main theorem is about iterated forcing for making the continuum larger than ℵ2. We present a generalization of [Sh:669] which deal with oracles for random, (also for other cases and generalities), by replacing ℵ1, ℵ2 by λ, λ + (starting with λ = λ <λ> ℵ1). Well, we demand absolute c.c.c. ..."
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Cited by 1 (1 self)
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Our main theorem is about iterated forcing for making the continuum larger than ℵ2. We present a generalization of [Sh:669] which deal with oracles for random, (also for other cases and generalities), by replacing ℵ1, ℵ2 by λ, λ + (starting with λ = λ <λ> ℵ1). Well, we demand absolute c.c.c. So we get, e.g. the continuum is λ + but we can get cov(meagre) = λ and we give some applications. As in [Sh:669], it is a “partial ” countable support iteration but it is c.c.c.
A PARALLEL TO THE NULL IDEAL FOR INACCESSIBLE λ
, 2012
"... Abstract. It is well known to generalize the meagre ideal replacing ℵ0 by a (regular) cardinal λ> ℵ0 and requiring the ideal to be λ +complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing ℵ0 by λ, so ..."
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Abstract. It is well known to generalize the meagre ideal replacing ℵ0 by a (regular) cardinal λ> ℵ0 and requiring the ideal to be λ +complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing ℵ0 by λ, so requiring it to be (< λ)complete. Of course, we would welcome additional properties generalizing the ones of the random real forcing. Returning to the ideal (instead of forcing) we may look at the Boolean Algebra of λBorel sets modulo the ideal. Surprisingly we get a positive = existence answer for λ a “mild ” large cardinal: the weakly compact one. We apply this to get consistency results on cardinal invariants for such λ’s. We shall deal with other cardinals more properties related forcing notions in a continuation. (1004) revision:20120524 modified:20120528