Independence results concerning Dedekindfinite sets

Monro, G. P.

doi:10.1017/s1446788700023521

Cited by 8 publications

(6 citation statements)

References 3 publications

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“…We show that a minor modification (in line with the observation) of the previous work of the author in [8] (which was exactly a partial solution that covered all ground model sets, but not necessarily all sets) we can in fact obtain a general framework for ∃∀ solutions, and we use it to prove that every partial order can be embedded into the cardinals of a model (extending [3] and [15] which obtained a local solution, and [6] where a global solution for ground model is shown). 1 In addition we prove that in this model every set can be the surjective image of a Dedekind-finite set (extending the local, and global for ground model solutions of [13]).…”

Section: Introductionmentioning

confidence: 80%

“…Most of the proofs were local, in the sense that for a given set we can build a specific extension of the universe where the axiom of choice fails and we have a certain witness for a certain failure of choice related to our set (e.g. have a Dedekindfinite set which maps onto that set [13]). But there were not that many successful attempts in constructing global results, namely extending the universe once, so that for every set the counterexample can be found in that extension.…”

Section: Introductionmentioning

confidence: 99%

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Iterated failures of choice

Karagila

2019

Preprint

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We make an important observation regarding iterated symmetric extensions which allows us to easily iterate certain counterexamples to the axiom of choice. We use this to improve previous results and to obtain a model where: (1) Every partially ordered set can be embedded into the cardinals; and(2) for every non-empty set X there is a set A which is the countable union of countable sets, and P(A) has a Dedekind-finite subset that can be mapped onto X. We also provide an outline for a construction of a model where every field admits a non-trivial vector space whose endomorphisms are all scalar multiplications. This method can be applied to similar "∀∃-∃∀ problems".

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Section: Introductionmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Iterated failures of choice

Karagila

2019

Preprint

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“…Fix a family Í Í ¾ Á of pairwise disjoint open sets of . Since, by the claim, the countable family has no choice function, it follows that has no partial choice (see [17] µ is a ccc compact metric space. We claim that does not embed in ¼ ½ ¼ .…”

Section: Questionmentioning

confidence: 99%

The failure of the axiom of choice implies unrest in the theory of Lindelöf metric spaces

Keremedis

2003

Mathematical Logic Qtrly

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In the realm of metric spaces the role of choice principles is investigated. Notation and terminologyDefinition 1 (i) The countable axiom of choice CAC (Form 8 in [7]) is the assertion: For every set ¾ of non-empty disjoint sets there exists a set consisting of one and only one element from each element of . (ii) CAC ¬Ò (Form 10 in [7]) is CAC restricted to families of disjoint finite sets. (iii) CAC(Ê) (Form 94 in [7]) is CAC restricted to families of subsets of the real line Ê.(iv) The countable union theorem CUC (Form 31 in [7]) is the statement: The countable union of countable sets is countable.(v) Form 368 is the statement: The set Ê of all denumerable´countably infiniteµ subsets of Ê has size Ê .Definition 2 Let´ Ìµ be a topological space.(i) is said to be Lindelöf iff every open cover Í of has a countable subcover Î.(ii) is said to be weakly Lindelöf iff every open cover Í has a countable subfamily Î such that Ë Î .(iii) has the ccc property iff every family of pairwise disjoint open subsets of is countable. (iv) A metric space´ µ is called preLindelöf iff for every ¼ space can be covered by countably many open discs of radius .(v) LMC stands for the proposition "Lindelöf metric spaces embed in compact metric spaces". In what follows we shall abbreviate "separable" by S, "second countable" by 2, "Lindelöf" by L, "weakly Lindelöf" by WL, "preLindelöf" by PL, "hereditarily Lindelöf" by "hL" and "hereditarily separable" by "hS". M(P, Q) stands for the statement: Every metric space´ µ which has the property P has also the property Q.

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“…In general, this final problem seems to be a difficult one to answer because, as shown by Monro in [Mon75], it is consistent to have a Dedekind-finite proper class:…”

Section: Now Definementioning

confidence: 99%

Taking Reinhardt's Power Away

Matthews¹

2020

Preprint

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We study the notion of non-trivial elementary embeddings j : V → V under the assumption that V satisfies ZFC without Power Set but with the Collection Scheme. We show that no such embedding can exist under the additional assumption that it is cofinal and either V crit(j ) is a set or that the Reflection Principle holds. We then study failures of instances of collection in symmetric submodels of class forcings.

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Independence results concerning Dedekindfinite sets

Cited by 8 publications

References 3 publications

Iterated failures of choice

Iterated failures of choice

The failure of the axiom of choice implies unrest in the theory of Lindelöf metric spaces

Taking Reinhardt's Power Away

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