Topics in Set Theory

Bekkali, Moulhime El

doi:10.1007/bfb0098398

Cited by 76 publications

(40 citation statements)

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“…As PFA implies MA + c = ℵ 2 (this was proved by Todorčević, see [5]), we know that there is an ℵ 2 -saturated ultrafilter u on ω, see [10]. Such an ultrafilter has the property that every ultrapower over it of any countable structure is ℵ 2 -saturated.…”

Section: Lemma 55 If There Is With Respect To the Mod Finite Ordermentioning

confidence: 87%

A universal continuum of weight $\aleph $

Dow

Hart

2000

Trans. Amer. Math. Soc.

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Abstract. We prove that every continuum of weight ℵ 1 is a continuous image of theČech-Stone-remainder R * of the real line. It follows that under CH the remainder of the half line [0, ∞) is universal among the continua of weight c -universal in the 'mapping onto' sense.We complement this result by showing that 1) under MA every continuum of weight less than c is a continuous image of R * , 2) in the Cohen model the long segment of length ω 2 + 1 is not a continuous image of R * , and 3) PFA implies that Iu is not a continuous image of R * , whenever u is a c-saturated ultrafilter.We also show that a universal continuum can be gotten from a c-saturated ultrafilter on ω, and that it is consistent that there is no universal continuum of weight c.

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Section: Lemma 55 If There Is With Respect To the Mod Finite Ordermentioning

confidence: 87%

A universal continuum of weight $\aleph $

Dow

Hart

2000

Trans. Amer. Math. Soc.

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“…14 It would be by different and elegant means that Todorcevic would show in 1990 (cf. [Bekkali, 1991]) that PFA already implies that 2 ℵ 0 = ℵ 2 .…”

Section: Reflecting Stationary Setsmentioning

confidence: 96%

“…Todorcevic (cf. [Bekkali, 1991]) formulated a strong reflection principle, one that also follows from MM but moreover has as a consequence that NS ω1 is ℵ 2 -saturated. Qi Feng and Jech [1998] also came up with a stronger reflection principle, but they soon showed it to be equivalent to Todorcevic's.…”

Section: Conjecture 15mentioning

confidence: 99%

Large Cardinals with Forcing

Kanamori¹

2012

Handbook of the History of Logic

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“…In fact, the two boil down to same, since Veličković and Todorčević proved in [34] and [7] that PFA implies c = ℵ 2 . Proper forcing has many applications, in the set theory of the reals, combinatorial properties of ω 1 and topology, algebra and analysis.…”

Section: Beyond CCCmentioning

confidence: 99%

Forcing Axioms, Finite Conditions and Some More

Džamonja

2013

Logic and Its Applications

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Abstract. We survey some classical and some recent results in the theory of forcing axioms, aiming to present recent breakthroughs and interest the reader in further developing the theory. The article is written for an audience of logicians and mathematicians not necessarily familiar with set theory. IntroductionWe shall work within the axioms of the Zermelo-Fraenkel set theory with Choice (ZFC). These axioms were introduced basically starting from 1908 and improving to a final version in the 1920s as an attempt to axiomatize the foundations of mathematics. There have been other such attempts at about the same time and later, but it is fair to say that for the purposes of much of modern mathematics the axioms of ZFC represent the accepted foundation (see [13] for a detailed discussion of foundational issues in set theory). Gödel's Incompleteness theorems [16] prove that for any consistent theory T which implies the Peano Axioms and whose axioms are presentable as a recursively enumerable set of sentences, so for any reasonable theory one would say, there is a sentence ϕ in the language of T such that T does not prove or disprove ϕ. In some sense the discussion of which axioms to use is made less interesting by these theorems, which can be interpreted as saying that a perfect choice of axioms does not exists. We therefore do like the most, we concentrate on the axioms that correctly model most of mathematics, and for the rest, we try to understand the limits and how we can improve them. For us ZFC is a basis for a foundation which in some circumstances can be extended to a larger set of axioms which provide an insight into various parts of mathematics. In here we concentrate on the forcing axioms (and their negations).

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Topics in Set Theory

Cited by 76 publications

References 0 publications

A universal continuum of weight $\aleph $

A universal continuum of weight $\aleph $

Large Cardinals with Forcing

Forcing Axioms, Finite Conditions and Some More

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