Displaying similar documents to “New axioms in set theory”

Axioms which imply GCH

Jan Mycielski (2003)

Fundamenta Mathematicae

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We propose some new set-theoretic axioms which imply the generalized continuum hypothesis, and we discuss some of their consequences.

Lusin sequences under CH and under Martin's Axiom

Uri Abraham, Saharon Shelah (2001)

Fundamenta Mathematicae

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Assuming the continuum hypothesis there is an inseparable sequence of length ω₁ that contains no Lusin subsequence, while if Martin's Axiom and ¬ CH are assumed then every inseparable sequence (of length ω₁) is a union of countably many Lusin subsequences.

Cardinality of the sets of all bijections, injections and surjections

Marcin Zieliński (2019)

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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The results of Zarzycki for the cardinality of the sets of all bijections, surjections, and injections are generalized to the case when the domains and codomains are infinite and different. The elementary proofs the cardinality of the sets of bijections and surjections are given within the framework of the Zermelo-Fraenkel set theory with the axiom of choice. The case of the set of all injections is considered in detail and more explicit an expression is obtained when the Generalized...

Set-theoretic constructions of two-point sets

Ben Chad, Robin Knight, Rolf Suabedissen (2009)

Fundamenta Mathematicae

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A two-point set is a subset of the plane which meets every line in exactly two points. By working in models of set theory other than ZFC, we demonstrate two new constructions of two-point sets. Our first construction shows that in ZFC + CH there exist two-point sets which are contained within the union of a countable collection of concentric circles. Our second construction shows that in certain models of ZF, we can show the existence of two-point sets without explicitly invoking the...

Large continuum, oracles

Saharon Shelah (2010)

Open Mathematics

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Our main theorem is about iterated forcing for making the continuum larger than ℵ2. We present a generalization of [2] which deal with oracles for random, (also for other cases and generalities), by replacing ℵ1,ℵ2 by λ, λ + (starting with λ = λ <λ > ℵ1). Well, we demand absolute c.c.c. So we get, e.g. the continuum is λ + but we can get cov(meagre) = λ and we give some applications. As in non-Cohen oracles [2], it is a “partial” countable support iteration but it is c.c.c. ...

Induced open projections and C*-smoothness

Włodzimierz J. Charatonik, Alejandro Illanes, Verónica Martínez-de-la-Vega (2013)

Colloquium Mathematicae

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We show that there exists a C*-smooth continuum X such that for every continuum Y the induced map C(f) is not open, where f: X × Y → X is the projection. This answers a question of Charatonik (2000).

On Multiset Ordering

Grzegorz Bancerek (2016)

Formalized Mathematics

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Formalization of a part of [11]. Unfortunately, not all is possible to be formalized. Namely, in the paper there is a mistake in the proof of Lemma 3. It states that there exists x ∈ M1 such that M1(x) > N1(x) and (∀y ∈ N1)x ⊀ y. It should be M1(x) ⩾ N1(x). Nevertheless we do not know whether x ∈ N1 or not and cannot prove the contradiction. In the article we referred to [8], [9] and [10].

Non-separating subcontinua of planar continua

D. Daniel, C. Islas, R. Leonel, E. D. Tymchatyn (2015)

Colloquium Mathematicae

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We revisit an old question of Knaster by demonstrating that each non-degenerate plane hereditarily unicoherent continuum X contains a proper, non-degenerate subcontinuum which does not separate X.

There is no complete axiom system for shuffle expressions

A. Szepietowski (2010)

RAIRO - Theoretical Informatics and Applications

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In this paper we show that neither the set of all valid equations between shuffle expressions nor the set of schemas of valid equations is recursively enumerable. Thus, neither of the sets can be recursively generated by any axiom system.