scholarly journals Приглашение в булевозначный анализ

2018 ◽  
Author(s):  
A.G. Kusraev ◽  
S.S. Kutateladze
Keyword(s):  
Continuum Hypothesis ◽  
Synthetic Methods ◽  
Mathematical Research ◽  
Analysis Model ◽  
Straight Line ◽  
Boolean Valued Analysis ◽  
Catch Up ◽  
Opening Up ◽  
Models Of Set Theory ◽  
The Continuum

This is a short invitation to the field of Boolean valued analysis. Model theory evaluates and counts truth and proof. The chase of truth not only leads us close to the truth we pursue but also enables us to nearly catch up with many other instances of truth which we were not aware nor even foresaw at the start of~the rally pursuit. That is what we have learned from Boolean valued models of set theory. These models stem from the famous works by Paul Cohen on the continuum hypothesis. They belong to logic and yield a~profusion of the surprising and unforeseen visualizations of the ingredients of mathematics. Many promising opportunities are open to modeling the powerful habits of reasoning and verification. Boolean valued analysis is a blending of analysis and Boolean valued models. Adaptation of the ideas of Boolean valued models to functional analysis projects among the most important directions of developing the synthetic methods of mathematics. This approach yields the new models of numbers, spaces, and types of equations. The content expands of all available theorems and algorithms. The whole methodology of mathematical research is enriched and renewed, opening up absolutely fantastic opportunities. We can now transform matrices into numbers, embed function spaces into a straight line, yet having still uncharted vast territories of new knowledge. The article advertised two books that crown our thought about and research into the field.

Power-like models of set theory

2001 ◽  
Vol 66 (4) ◽  
pp. 1766-1782 ◽  
Author(s):  
Ali Enayat
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Mahlo Cardinal ◽  
Uncountable Cardinal ◽  
Consistent Extension ◽  
Elementary Submodel ◽  
Finite Language ◽  
The Universe ◽  
Models Of Set Theory ◽  
The Continuum

Abstract.A model = (M. E, …) of Zermelo-Fraenkel set theory ZF is said to be 0-like. where E interprets ∈ and θ is an uncountable cardinal, if ∣M∣ = θ but ∣{b ∈ M: bEa}∣ < 0 for each a ∈ M, An immediate corollary of the classical theorem of Keisler and Morley on elementary end extensions of models of set theory is that every consistent extension of ZF has an ℵ1-like model. Coupled with Chang's two cardinal theorem this implies that if θ is a regular cardinal 0 such that 2<0 = 0 then every consistent extension of ZF also has a 0+-like model. In particular, in the presence of the continuum hypothesis every consistent extension of ZF has an ℵ2-like model. Here we prove:Theorem A. If 0 has the tree property then the following are equivalent for any completion T of ZFC:(i) T has a 0-like model.(ii) Ф ⊆ T. where Ф is the recursive set of axioms {∃κ (κ is n-Mahlo and “Vκis a Σn-elementary submodel of the universe”): n ∈ ω}.(iii) T has a λ-like model for every uncountable cardinal λ.Theorem B. The following are equiconsistent over ZFC:(i) “There exists an ω-Mahlo cardinal”.(ii) “For every finite language , all ℵ2-like models of ZFC() satisfy the schemeФ().

Martin's axiom and the continuum

1995 ◽  
Vol 60 (2) ◽  
pp. 374-391 ◽  
Author(s):  
Haim Judah ◽  
Andrzej Rosłanowski
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Point Of View ◽  
Georg Cantor ◽  
Mathematical Research ◽  
Kurt Gödel ◽  
Uncountable Cardinal ◽  
Image Position ◽  
Infinite Set ◽  
The Continuum

Since Georg Cantor discovered set theory the main problem in this area of mathematical research has been to discover what is the size of the continuum. The continuum hypothesis (CH) says that every infinite set of reals either has the same cardinality as the set of all reals or has the cardinality of the set of natural numbers, namelyIn 1939 Kurt Gödel discovered the Constructible Universe and proved that CH holds in it. In the early sixties Paul Cohen proved that every universe of set theory can be extended to a bigger universe of set theory where CH fails. Moreover, given any reasonable cardinal κ, it is possible to build a model where the continuum size is κ. The new technique discovered by Cohen is called forcing and is being used successfully in other branches of mathematics (analysis, algebra, graph theory, etc.).In the light of these two stupendous works the experts (especially the platonists) were forced to conclude that from the point of view of the classical axiomatization of set theory (called ZFC) it is impossible to give any answer to the continuum size problem: everything is possible!In private communications Gödel suggested that the continuum size from a platonistic point of view should be ω2, the second uncountable cardinal. As this is not provable in ZFC, Gödel suggested that a new axiom should be added to ZFC to decide that the cardinality of the continuum is ω2.

scholarly journals Several results on compact metrizable spaces in $$\mathbf {ZF}$$

2021 ◽  
Author(s):  
Kyriakos Keremedis ◽  
Eleftherios Tachtsis ◽  
Eliza Wajch
Keyword(s):  
Continuum Hypothesis ◽  
Compact Spaces ◽  
Power Set ◽  
Compact Metrizable Space ◽  
Permutation Models ◽  
Independence Results ◽  
The Axiom Of Choice ◽  
Metrizable Spaces ◽  
Transfer Theorems ◽  
The Continuum

AbstractIn the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in $$\mathbf {ZF}$$ ZF , some are shown to be independent of $$\mathbf {ZF}$$ ZF . For independence results, distinct models of $$\mathbf {ZF}$$ ZF and permutation models of $$\mathbf {ZFA}$$ ZFA with transfer theorems of Pincus are applied. New symmetric models of $$\mathbf {ZF}$$ ZF are constructed in each of which the power set of $$\mathbb {R}$$ R is well-orderable, the Continuum Hypothesis is satisfied but a denumerable family of non-empty finite sets can fail to have a choice function, and a compact metrizable space need not be embeddable into the Tychonoff cube $$[0, 1]^{\mathbb {R}}$$ [ 0 , 1 ] R .

Sign in / Sign up

Export Citation Format

Share Document