Hamel bases and well–ordering the continuum

Beriashvili, Mariam; Schindler, Ralf; Wu, Liuzhen; Yu, Liang

doi:10.1090/proc/14010

Cited by 9 publications

(22 citation statements)

References 3 publications

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“…We assume that V satisfies ZFC, 1 and we take P to be Add(ω, ω). Our group of automorphisms is given by the group of finitary permutations of ω acting on P in the natural way 2 And finally, F is the filter of subgroups generated by 3 If fix(E) ⊆ sym(ẋ), we say that E is a support forẋ.…”

Section: Cohen's First Modelmentioning

confidence: 99%

“…Over the years, we see time and time again how rich and interesting the theory of Cohen's first model is. Recently, for example, Beriashvili et al proved in [1] that in Cohen's first model there is a Hamel basis for the real numbers as a vector space over the rational numbers.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, for example, Beriashvili et al . proved in [ 1 ] that in Cohen’s first model there is a Hamel basis for the real numbers as a vector space over the rational numbers.…”

Section: Introductionmentioning

confidence: 99%

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How to have more things by forgetting how to count them

Karagila

Schlicht

2020

Proc. R. Soc. A.

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Cohen’s first model is a model of Zermelo–Fraenkel set theory in which there is a Dedekind-finite set of real numbers, and it is perhaps the most famous model where the Axiom of Choice fails. We force over this model to add a function from this Dedekind-finite set to some infinite ordinal κ . In the case that we force the function to be injective, it turns out that the resulting model is the same as adding κ Cohen reals to the ground model, and that we have just added an enumeration of the canonical Dedekind-finite set. In the case where the function is merely surjective it turns out that we do not add any reals, sets of ordinals, or collapse any Dedekind-finite sets. This motivates the question if there is any combinatorial condition on a Dedekind-finite set A which characterises when a forcing will preserve its Dedekind-finiteness or not add new sets of ordinals. We answer this question in the case of ‘Adding a Cohen subset’ by presenting a varied list of conditions each equivalent to the preservation of Dedekind-finiteness. For example, 2 A is extremally disconnected, or [ A ] < ω is Dedekind-finite.

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Section: Cohen's First Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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How to have more things by forgetting how to count them

Karagila

Schlicht

2020

Proc. R. Soc. A.

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“…This last part raises an interesting question. In [4] the authors show that in Cohen's model there is a Hamel basis for R over Q. Cohen's model is famous of having a Dedekind-finite set of reals, and therefore DC fails quite badly. However, it is also very different from Feferman's construction of a model satisfying V = L(R) where the Boolean Prime Ideal theorem fails, in that the set of Cohen reals is in Cohen's model but not in Feferman's model.…”

Section: Sets Of Reals and Dependent Choicementioning

confidence: 99%

Preserving Dependent Choice

Karagila

2019

Bull. Polish Acad. Sci. Math.

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Definable Hamel bases and $AC_ω(R)$

Kanovei,

Schindler

2019

Preprint

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There is a model of ZF with a ∆ 1 3 definable Hamel bases in which ACω(R) fails.Answering a question from [13, p. 433] it was shown in [2] that there is a Hamel basis in the Cohen-Halpern-Lévy model. In this paper we show that in a variant of this model, there is a projective, in fact ∆ 1 3 , Hamel basis. Throughout this paper, by a Hamel basis we always mean a basis for R, construed as a vector space over Q. We denote by E 0 the Vitali equivalence relation, xE 0 y iff x − y ∈ Q for x, y ∈ R. We also write [x] E0 = {y : yE 0 x} for the E 0 -equivalence class of x. A transversal for the set of all E 0 -equivalence classes picks exactly one member from each [x] E0 . The range of any such transversal is also called a Vitali set. If we identify R with the Cantor space ω 2, then xE 0 y iff {n :at most countable for every null set N ⊂ R ("null" in the sense of Lebesgue measure). A set B ⊂ R is a Bernstein set iff B ∩ P = ∅ = P \ B for every perfect set P ⊂ R. A Burstin basis is a Hamel basis which is also a Bernstein set. It is easy to see that B ⊂ R is a Burstin basis iff B is a Hamel basis and B ∩ P = ∅ for every perfectBy AC ω (R) we mean the statement that for all sequences (A n : n < ω) such that ∅ = A n ⊂ R for all n < ω there is some choice function f : ω → R, i.e., f (n) ∈ A n for all n < ω.D. Pincus and K. Prikry study the Cohen-Halpern-Lévy model H in [13]. The model H is obtained by adding a countable set of Cohen reals (say over L) without adding their enumeration; H does not satisfy AC ω (R). It is shown in [13] that there is a Luzin set in H, so that in ZF, the existence of a Luzin set does not even imply AC ω (R). [2, Theorems 1.7 and 2.1] show that in H there is a Bernstein set as well as a Hamel basis. As in ZF the existence of a Hamel basis implies the existence of a Vitali set, the latter also reproves Feferman's result (see [13]) according to which there is a Vitali set in H.

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Hamel bases and well–ordering the continuum

Cited by 9 publications

References 3 publications

How to have more things by forgetting how to count them

How to have more things by forgetting how to count them

Preserving Dependent Choice

Definable Hamel bases and $AC_ω(R)$

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