Non-constructive Properties of the Real Numbers

Howard, Paul E.; Keremedis, Kyriakos; Rubin, Jean E.; Stanley, Adrienne; Tatchtsis, Eleftherios

doi:10.1002/1521-3870(200108)47:3<423::aid-malq423>3.0.co;2-0

Cited by 16 publications

(15 citation statements)

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“…By mimicking the proof of the claim we can also show that for all n, m ∈ k, if B n ∩ B m = ∅, then C n = C m , where C n , C m are given by (1). Define now a relation ∼ on k by requiring for all n, m ∈ k, n∼m iff C n = C m .…”

Section: (I) ⇒ (Iii) Fix a And B As In (Iii) For Each I ∈ K Letmentioning

confidence: 88%

Choice principles for special subsets of the real line

Keremedis¹,

Tachtsis

2003

Mathematical Logic Qtrly

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Section: (I) ⇒ (Iii) Fix a And B As In (Iii) For Each I ∈ K Letmentioning

confidence: 88%

Choice principles for special subsets of the real line

Keremedis¹,

Tachtsis

2003

Mathematical Logic Qtrly

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“…(10) In [6] we showed that Form 203 implies Form 170, and by (8) we deduce that Form 203 implies Form 169. The implication "Form 203 ⇒ Form 212" has been proved in [10].…”

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confidence: 90%

“…(6) In M5(ℵ), CC(R) is true, whereas Form 169 fails, see [7]. The second assertion of (6) has been proved in [10,Proposition 8]. …”

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confidence: 93%

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Properties of the real line and weak forms of the Axiom of Choice

Cruz

Hall

Howard

et al. 2005

MLQ - Math. Log. Quart.

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“…£ Example 1 demonstrates explicitly the fact that the most fundamental questions concerning Lindelöf (compact) metric spaces can be undecidable from the Zermelo-Frankel axioms of set theory alone. More examples of the counter-intuitive behavior of metric spaces, in absence of AC, can be found in [2,6,3,4,14,12,13,15,8,9,10].…”

Section: Introductionmentioning

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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Non-constructive Properties of the Real Numbers

Abstract: We study the relationship between various properties of the real numbers and weak choice principles.Mathematics Subject Classification: 03E25, 04A25.

Cited by 16 publications

References 6 publications

Choice principles for special subsets of the real line

Choice principles for special subsets of the real line

Properties of the real line and weak forms of the Axiom of Choice

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

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