The Notion of Forcing

Halbeisen, Lorenz

doi:10.1007/978-3-319-60231-8_15

Cited by 10 publications

(20 citation statements)

References 7 publications

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“…In this section, we establish some consistency results by the method of permutation models (cf. [2,Chapter 8]). Permutation models are not models of 𝖹𝖥; they are models of 𝖹𝖥𝖠 (the Zermelo-Fraenkel set theory with atoms).…”

Section: Consistency Resultsmentioning

confidence: 99%

The power set and the set of permutations with finitely many non‐fixed points of a set

Shen

2023

Mathematical Logic Qtrly

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For a cardinal a$\mathfrak {a}$, we write scriptSfin(a)$\operatorname{\mathcal {S}_{\text{fin}}}(\mathfrak {a})$ for the cardinality of the set of permutations with finitely many non‐fixed points of a set which is of cardinality a$\mathfrak {a}$. We investigate the relationships between 2fraktura$2^\mathfrak {a}$ and scriptSfin(a)$\operatorname{\mathcal {S}_{\text{fin}}}(\mathfrak {a})$ for an arbitrary infinite cardinal a$\mathfrak {a}$ in ZF$\mathsf {ZF}$ (without the axiom of choice). It is proved in ZF$\mathsf {ZF}$ that 2a≠scriptSfin(a)$2^\mathfrak {a}\ne \operatorname{\mathcal {S}_{\text{fin}}}(\mathfrak {a})$ for all infinite cardinals a$\mathfrak {a}$, and we show that this is the best possible result.

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Section: Consistency Resultsmentioning

confidence: 99%

The power set and the set of permutations with finitely many non‐fixed points of a set

Shen

2023

Mathematical Logic Qtrly

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“…Fact 2 (Folklore, see [Bl], see also [Ha,Proposition 22.13] for Cohen forcing). After adding at least ω 1 Cohen or random reals to a model of ZFC, s = ω 1 .…”

Section: Preliminariesmentioning

confidence: 99%

Base matrices of various heights

Brendle

2023

Can. Math. Bull.

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“…More information about these cardinals can be found in e.g. [2,1,5,10]. Let M and N denote the ideals of meager and null sets respectively on 2 ω (or any other perfect Polish space, it does not matter for our purposes).…”

Section: Introductionmentioning

confidence: 99%

“…Most of our notation is standard, conforming to e.g. [2,10,13,16]. To begin we set some vocabulary for trees.…”

Section: Introductionmentioning

confidence: 99%

The special tree number

Switzer

2023

Fund. Math.

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Define the special tree number, denoted st, to be the least size of a tree of height ω1 which is neither special nor has a cofinal branch. This cardinal had previously been studied in the context of fragments of MA, but in this paper we look at its relation to other, more typical, cardinal characteristics. Classical facts imply that ℵ1 ≤ st ≤ 2 ℵ 0 , under Martin's Axiom st = 2 ℵ 0 , and that st = ℵ1 is consistent with MA(Knaster)+2 ℵ 0 = κ for any regular κ, thus the value of st is not decided by ZFC and in fact can be strictly below essentially all well studied cardinal characteristics. We show that conversely it is consistent that st = 2 ℵ 0 = κ for any κ of uncountable cofinality, while non(M) = a = s = g = ℵ1.In particular, st is independent of the left hand side of Cichoń's diagram, amongst other things. The proof involves an in-depth study of the standard ccc forcing notion to specialize (wide) Aronszajn trees, which may be of independent interest.

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The Notion of Forcing

Cited by 10 publications

References 7 publications

The power set and the set of permutations with finitely many non‐fixed points of a set

The power set and the set of permutations with finitely many non‐fixed points of a set

Base matrices of various heights

The special tree number

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