A model of set-theory in which every set of reals is Lebesgue measurable

Solovay, Robert M.

doi:10.1142/9789812564894_0025

Cited by 161 publications

(258 citation statements)

References 0 publications

Supporting

Mentioning

251

Contrasting

Unclassified

Order By: Relevance

“…On the other hand consider a model of set theory in which the axiom of choice fails, Solovay's (1970) model, or if every set has the property of Baire say. Then every function R → R is continuous on a set whose complement is meager; if it is also a group homomorphism, it will be continuous everywhere.…”

Section: The Effectiveness Of Gleason's Theoremmentioning

confidence: 99%

Generalizations of Kochen and Specker's theorem and the effectiveness of Gleason's theorem

Hrushovski

Pitowsky

2004

Studies in History and Philosophy of Science Part B: Studies in

View full text Add to dashboard Cite

Abstract. Kochen and Specker's theorem can be seen as a consequence of Gleason's theorem and logical compactness. Similar compactness arguments lead to stronger results about finite sets of rays in Hilbert space, which we also prove by a direct construction. Finally, we demonstrate that Gleason's theorem itself has a constructive proof, based on a generic, finite, effectively generated set of rays, on which every quantum state can be approximated.

show abstract

Section: The Effectiveness Of Gleason's Theoremmentioning

confidence: 99%

Generalizations of Kochen and Specker's theorem and the effectiveness of Gleason's theorem

Hrushovski

Pitowsky

2004

Studies in History and Philosophy of Science Part B: Studies in

View full text Add to dashboard Cite

show abstract

“…Solovay [17] showed that it is consistent with ZF plus dependent choice that all subsets of P(ω) are measurable and have the Baire property. His proof assumed the consistency of ZFC plus the existence of an inaccessible cardinal, but Shelah [15] eliminated the need for the inaccessible in the case of the Baire property.…”

Section: Corollary 29 An Infinitary Voting Rule On ω Is Not a Borel mentioning

confidence: 86%

“…We begin by showing that, as with ultrafilters, we cannot hope to establish the existence of voting rules in general Boolean algebras, or even in the specific case of P(ω)/fin, without using the axiom of choice. The proof is essentially the same as the standard proof of the corresponding result for ultrafilters, combining results from [16,17,15]. .…”

Section: Axiom Of Choicementioning

confidence: 96%

Voting rules for infinite sets and Boolean algebras

Blass¹

2007

Advances in Logic

View full text Add to dashboard Cite

Abstract. A voting rule in a Boolean algebra B is an upward closed subset that contains, for each element x ∈ B, exactly one of x and ¬x. We study several aspects of voting rules, with special attention to their relationship with ultrafilters. In particular, we study the set-theoretic hypothesis that all voting rules in the Boolean algebra of subsets of the natural numbers modulo finite sets are nearly ultrafilters. We define the notion of support of a voting rule and use it to describe voting rules that are, in a sense, as different as possible from ultrafilters. Finally, we consider how much of the axiom of choice is needed to guarantee the existence of voting rules.

show abstract

“…While it is possible to trade the axiom of choice for measurability of all sets [25], the accepted norm is to keep choice and deal with the phenomenon of nonmeasurability. A surprising alternative was proposed by Alex Simpson [23]: if we use locales instead of spaces, we can have both, the axiom of choice and an isometry-invariant measure on all sublocales of R n , which agrees with the Lebesgue measure on the measurable sets.…”

Section: Definition 56mentioning

confidence: 99%

Five stages of accepting constructive mathematics

Bauer

2016

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. On the odd day, a mathematician might wonder what constructive mathematics is all about. They may have heard arguments in favor of constructivism but are not at all convinced by them, and in any case they may care little about philosophy. A typical introductory text about constructivism spends a great deal of time explaining the principles and contains only trivial mathematics, while advanced constructive texts are impenetrable, like all unfamiliar mathematics. How then can a mathematician find out what constructive mathematics feels like? What new and relevant ideas does constructive mathematics have to offer, if any? I shall attempt to answer these questions.From a psychological point of view, learning constructive mathematics is agonizing, for it requires one to first unlearn certain deeply ingrained intuitions and habits acquired during classical mathematical training. In her book On Death and Dying [17] psychologist Elisabeth Kübler-Ross identified five stages through which people reach acceptance of life's traumatizing events: denial, anger, bargaining, depression, and acceptance. We shall follow her path.

show abstract

A model of set-theory in which every set of reals is Lebesgue measurable

Cited by 161 publications

References 0 publications

Generalizations of Kochen and Specker's theorem and the effectiveness of Gleason's theorem

Generalizations of Kochen and Specker's theorem and the effectiveness of Gleason's theorem

Voting rules for infinite sets and Boolean algebras

Five stages of accepting constructive mathematics

Contact Info

Product

Resources

About