Variants of R<scp>ADO'S</scp> Selection Lemma and their Applications

Rav, Yehuda

doi:10.1002/mana.19770790112

Cited by 27 publications

(8 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The statements

sans-serifUF

(ω) and

sans-serifBPI

(ω) were introduced in [, Problem 5.24] and , respectively. Clearly,

BPI (ω)

→

UF (ω)

is true in

sans-serifZF

, but the status of the implication

UF (ω) \to BPI (ω)

is indicated as unknown in and .…”

Section: Introduction and Some Preliminary Resultsmentioning

confidence: 99%

The existence of free ultrafilters on ω does not imply the extension of filters on ω to ultrafilters

Hall

Keremedis

Tachtsis

2013

Mathematical Logic Qtrly

View full text Add to dashboard Cite

Let X be an infinite set and let BPI(X) and UF(X) denote the propositions “every filter on X can be extended to an ultrafilter” and “X has a free ultrafilter”, respectively. We denote by S(X) the Stone space of the Boolean algebra of all subsets of X. We show: For every well‐ordered cardinal number ℵ, sans-serifUF(ℵ) iff sans-serifUF(2ℵ). UF(ω) iff “2ω is a continuous image of S(ω)” iff “S(ω) has a free open ultrafilter ” iff “every countably infinite subset of S(ω) has a limit point”. BPI(ω) implies “every open filter on 2ω extends to an open ultrafilter” implies “2ωhas an open ultrafilter” implies UF(ω). It is relatively consistent with sans-serifZF that sans-serifUF(ω) holds, whereas sans-serifBPI(ω) fails. In particular, none of the statements given in (2) implies sans-serifBPI(ω).

show abstract

“…The statements

sans-serifUF

(ω) and

sans-serifBPI

(ω) were introduced in [, Problem 5.24] and , respectively. Clearly,

BPI (ω)

→

UF (ω)

is true in

sans-serifZF

, but the status of the implication

UF (ω) \to BPI (ω)

is indicated as unknown in and .…”

Section: Introduction and Some Preliminary Resultsmentioning

confidence: 99%

The existence of free ultrafilters on ω does not imply the extension of filters on ω to ultrafilters

Hall

Keremedis

Tachtsis

2013

Mathematical Logic Qtrly

View full text Add to dashboard Cite

show abstract

“…By the one-to-one correspondence between Boolean algebras, Boolean lattices and Boolean rings, the Boolean Prime Ideal Theorem is a common specialization of (4) and (5). Moreover, it is known that each of the previous prime ideal theorems is equivalent to the Ultrafilter Principle (see [3,6,23,24,26]). As a consequence, the representation of Boolean algebras and distributive lattices as subdirect products of two-element chains, but also Stone's representation of bounded distributive lattices by compact -open sets in sober spaces, as well as their representation by clopen lower sets in Priestley spaces (see [22]), are equivalent to UP.…”

Section: (Cdc) the Principle Of Countable Dependent Choicesmentioning

confidence: 99%

Prime Ideal Theory for General Algebras

Erné

2000

Papers in Honour of Bernhard Banaschewski

View full text Add to dashboard Cite

We introduce ideals, radicals and prime ideals in arbitrary algebras with at least one binary operation, and we show that various separation lemmas and prime ideal theorems are special instances of one general theorem which, in tum, is equivalent to the Boolean Prime Ideal Theorem (or Ultrafilter Principle). (2000): Primary: 06F05, 08A30, 20M12, 20N02; Secondary: 03G05, 16030, 16N80, 17A65. Mathematics Subject Classifications

show abstract

“…There are other such non-constructive principles that are perhaps not so well known as those given above. Here is a short list (see [12], form 14, for a much longer list; additional references are [2], [3], [13], [18], and [20]). The logical status of these ideas is not trivial.…”

Section: Introductionmentioning

confidence: 99%

Restricted versions of the Tukey-Teichmüller theorem that are equivalent to the Boolean prime ideal theorem

Hodel

2004

Arch. Math. Logic

View full text Add to dashboard Cite

Dedicated to W.W. Comfort on the occasion of his seventieth birthday. AbstractWe formulate a restricted version of the Tukey-Teichmüller Theorem that we denote by (rTT). We then prove that (rTT) and (BPI) are equivalent in ZF and that (rTT) applies rather naturally to several equivalent forms of (BPI): Alexander Subbase Theorem, Stone Representation Theorem, Model Existence and Compactness Theorems for propositional and first-order logic. We also give two variations of (rTT) that we denote by (rTT) + and (rTT) ++ ; each is equivalent to (rTT) in ZF. The variation (rTT) ++ applies rather naturally to various Selection Lemmas due to Cowen, Engeler, and Rado. IntroductionLet ZF denote Zermelo-Fraenkel set theory. It is well known that the axiom of choice (AC) is independent of ZF. When we add AC to ZF we obtain the most widely used axiom system for classical mathematics, namely ZFC = ZF + AC.It is also well known that there are many theorems of ZFC that are in fact equivalent forms of AC. For example, we have the following short list (see [12], [13], and [19] for much longer lists): In this paper we are especially interested in (TT). Recall that a collection A of subsets of a set X has finite character if for every A ⊆ X, A ∈ A if and only if every finite subset of A is in A. Note that ∅ ∈ A (assuming that A is non-empty). There are two versions of the Tukey-Teichmüller Theorem; however, in ZF they are equivalent to each other and also to AC.TT (weak form) Let X be a set and let A be a non-empty collection of subsets of X with finite character. Then A has a maximal element (with respect to ⊆).TT (strong form) Let X be a set and let A be a non-empty collection of subsets of X with finite character. Then for each A ∈ A, there exists B ∈ A such that A ⊆ B and B is maximal.The Tukey-Teichmüller Theorem seems tailor-made for certain applications. For example, it can be used to prove that every vector space has a basis by simply noting that the property of being linearly independent has finite character. Another nice application is a proof of the Alexander Subbase Theorem (see [14] or [7]).The axiom of choice and its equivalent forms are non-constructive principles; they assert the existence of a set without giving instructions on how to construct the set. There are other such non-constructive principles that are perhaps not so well known as those given above. Here is a short list (see The logical status of these ideas is not trivial. The easy part is:The hard part states:• AC cannot be proved from ZF + BPI (Halpern and Lévy [10]);• BPI cannot be proved from ZF (S. Feferman [8]).In section 2 we formulate a version of the Tukey-Teichmüller Theorem (TT) that is equivalent to (BPI) rather than to (AC). We then use this new 2 non-constructive axiom, denoted (rTT) (for restricted Tukey-Teichmüller), to give direct proofs of the Alexander Subbase Theorem and the Stone Representation Theorem. In section 3 we show directly that (rTT) and (BPI) are equivalent in ZF. Finally, in section 4 we use (rTT) to derive various theorems...

show abstract

Variants of RADO'S Selection Lemma and their Applications

Cited by 27 publications

References 33 publications

The existence of free ultrafilters on ω does not imply the extension of filters on ω to ultrafilters

The existence of free ultrafilters on ω does not imply the extension of filters on ω to ultrafilters

Prime Ideal Theory for General Algebras

Restricted versions of the Tukey-Teichmüller theorem that are equivalent to the Boolean prime ideal theorem

Contact Info

Product

Resources

About