The continuum hypothesis in intuitionism

1981 ◽  
Vol 46 (1) ◽  
pp. 121-136 ◽  
Author(s):  
W. Gielen ◽  
H. de Swart ◽  
W. Veldman
Keyword(s):  
Set Theory ◽  
Real Line ◽  
Classical Logic ◽  
Closed Subset ◽  
Continuum Hypothesis ◽  
The Real ◽  
The Core ◽  
The Continuum

Although Brouwer became famous for his vehement attacks upon classical logic and set theory, his work did not develop in a vacuum and strongly depended on that of Cantor.His mind bent on shifting aside nonconstructive arguments, he tried to rebuild Cantor's edifice along new, intuitionistic lines. The continuum hypothesis, lying at the core of set theory, also confronted Brouwer, and he had to face the farthest conclusion Cantor had been able to reach in trying to solve it: every nondenumerable closed subset of the real line has the power of the continuum.Brouwer's thinking about it seems to have been subject to some development. In 1914 we hear him saying: “Wir sahen oben dass das Cantorsche Haupttheorem für den Intuitionisten keines Beweises bedarf” (“As we saw above, for us, being intuitionists, Cantor's Main Theorem does not need a proof”) [3]. Nevertheless, five years later, he publishes an essay: Theorie der Punktmengen, which might be described as an attempt to reconstruct Cantor's reasonings in detail [4].This attempt was not entirely successful, as Brouwer comes to admit in 1952, probably having lost, now, some of his youthful rashness [10]. So the question of what the constructive content of Cantor's Main Theorem is, still awaits an answer.We do not think the answer we will give can be considered a conclusive one, but, in any case, it is a beginning.

Set theoretic naturalism

1996 ◽  
Vol 61 (2) ◽  
pp. 490-514 ◽  
Author(s):  
Penelope Maddy
Keyword(s):  
Set Theory ◽  
Classical Logic ◽  
Continuum Hypothesis ◽  
Philosophical Analysis ◽  
Central Question ◽  
The Status ◽  
Classical Mathematics ◽  
The Continuum

My aim in this paper is to propose what seems to me a distinctive approach to set theoretic methodology. By ‘methodology’ I mean the study of the actual methods used by practitioners, the study of how these methods might be justified or reformed or extended. So, for example, when the intuitionist's philosophical analysis recommends a wholesale revision of the methods of proof used in classical mathematics, this is a piece of reformist methodology. In contrast with the intuitionist, I will focus more narrowly on the methods of contemporary set theory, and, more importantly, I will certainly recommend no sweeping reforms. Rather, I begin from the assumption that the methodologist's job is to account for set theory as it is practiced, not as some philosophy would have it be. This credo lies at the very heart of the so-called ‘naturalism’ to be described here.A philosopher looking at set theoretic practice from the outside, so to speak, might notice any number of interesting methodological questions, beginning with the intuitionist's ‘why use classical logic?’, but this sort of question is not a live issue for most practicing set theorists. One central question on which the philosopher's and the practitioner's interests converge is this: what is the status of independent statements like the continuum hypothesis (CH)? A number of large questions arise in its wake: what criteria should guide the search for new axioms? For that matter, what reasons support our adoption of the old axioms?

scholarly journals Applying Set Theory E88

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 329
Author(s):  
Saharon Shelah
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Real Line ◽  
Cantor Sets ◽  
General Topology ◽  
The Real ◽  
Collectionwise Hausdorff ◽  
Monadic Logic

We prove some results in set theory as applied to general topology and model theory. In particular, we study ℵ1-collectionwise Hausdorff, Chang Conjecture for logics with Malitz-Magidor quantifiers and monadic logic of the real line by odd/even Cantor sets.

A note on the Borel types of some small sets

2018 ◽  
Vol 25 (3) ◽  
pp. 419-425
Author(s):  
Alexander Kharazishvili
Keyword(s):  
Real Line ◽  
The Real ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
The Continuum

AbstractThe Borel types of some classical small subsets of the real line are considered. In particular, under Martin’s axiom it is shown that there are at least {{\mathbf{c}}^{+}} pairwise incomparable Borel types of generalized Luzin sets (resp. of generalized Sierpiński sets), where {{\mathbf{c}}} stands for the cardinality of the continuum.

Set theoretic properties of Loeb measure

1990 ◽  
Vol 55 (3) ◽  
pp. 1022-1036 ◽  
Author(s):  
Arnold W. Miller
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Measure Zero ◽  
Continuum Hypothesis ◽  
Zero Set ◽  
Zero Sets ◽  
Martin’S Axiom ◽  
Basic Set ◽  
Nonstandard Models ◽  
The Continuum

AbstractIn this paper we ask the question: to what extent do basic set theoretic properties of Loeb measure depend on the nonstandard universe and on properties of the model of set theory in which it lies? We show that, assuming Martin's axiom and κ-saturation, the smallest cover by Loeb measure zero sets must have cardinality less than κ. In contrast to this we show that the additivity of Loeb measure cannot be greater than ω1. Define cof(H) as the smallest cardinality of a family of Loeb measure zero sets which cover every other Loeb measure zero set. We show that card(⌊log2(H)⌋) ≤ cof (H) ≤ card(2H), where card is the external cardinality. We answer a question of Paris and Mills concerning cuts in nonstandard models of number theory. We also present a pair of nonstandard universes M ≼ N and hyperfinite integer H ∈ M such that H is not enlarged by N, 2H contains new elements, but every new subset of H has Loeb measure zero. We show that it is consistent that there exists a Sierpiński set in the reals but no Loeb-Sierpiński set in any nonstandard universe. We also show that it is consistent with the failure of the continuum hypothesis that Loeb-Sierpiński sets can exist in some nonstandard universes and even in an ultrapower of a standard universe.

scholarly journals On Arbitrary sets andZFC

2011 ◽  
Vol 17 (3) ◽  
pp. 361-393 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Real Numbers ◽  
The Real ◽  
Formal Systems ◽  
Infinite Sets ◽  
The Axiom Of Choice ◽  
The Continuum

AbstractSet theory deals with the most fundamental existence questions in mathematics-questions which affect other areas of mathematics, from the real numbers to structures of all kinds, but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the so-called “arbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels ofquasi-combinatorialismorcombinatorial maximality. After explaining what is meant by definability and by “arbitrariness,” a first historical part discusses the strong motives why set theory was conceived as a theory of arbitrary sets, emphasizing connections with analysis and particularly with the continuum of real numbers. Judged from this perspective, the axiom of choice stands out as a most central and natural set-theoretic principle (in the sense of quasi-combinatorialism). A second part starts by considering the potential mismatch between the formal systems of mathematics and their motivating conceptions, and proceeds to offer an elementary discussion of how far the Zermelo–Fraenkel system goes in laying out principles that capture the idea of “arbitrary sets”. We argue that the theory is rather poor in this respect.

Forcing for the impredicative theory of classes

1972 ◽  
Vol 37 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Rolando Chuaqui
Keyword(s):  
General Theory ◽  
Set Theory ◽  
Continuum Hypothesis ◽  
Reflection Principle ◽  
Inaccessible Cardinal ◽  
Consistency Proof ◽  
Suitable Form ◽  
Definition Of ◽  
Class Construction ◽  
The Continuum

The purpose of this work is to formulate a general theory of forcing with classes and to solve some of the consistency and independence problems for the impredicative theory of classes, that is, the set theory that uses the full schema of class construction, including formulas with quantification over proper classes. This theory is in principle due to A. Morse [9]. The version I am using is based on axioms by A. Tarski and is essentially the same as that presented in [6, pp. 250–281] and [10, pp. 2–11]. For a detailed exposition the reader is referred there. This theory will be referred to as .The reflection principle (see [8]), valid for other forms of set theory, is not provable in . Some form of the reflection principle is essential for the proofs in the original version of forcing introduced by Cohen [2] and the version introduced by Mostowski [10]. The same seems to be true for the Boolean valued models methods due to Scott and Solovay [12]. The only suitable form of forcing for found in the literature is the version that appears in Shoenfield [14]. I believe Vopěnka's methods [15] would also be applicable. The definition of forcing given in the present paper is basically derived from Shoenfield's definition. Shoenfield, however, worked in Zermelo-Fraenkel set theory.I do not know of any proof of the consistency of the continuum hypothesis with assuming only that is consistent. However, if one assumes the existence of an inaccessible cardinal, it is easy to extend Gödel's consistency proof [4] of the axiom of constructibility to .

Letter Games: a metamathematical taster

2016 ◽  
Vol 100 (549) ◽  
pp. 442-449
Author(s):  
A. C. Paseau
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Incompleteness Theorem ◽  
Mathematical Study ◽  
Mathematical System ◽  
Kurt Gödel ◽  
Standard Set ◽  
The Continuum ◽  
Simplified Form

Metamathematics is the mathematical study of mathematics itself. Two of its most famous theorems were proved by Kurt Gödel in 1931. In a simplified form, Gödel's first incompleteness theorem states that no reasonable mathematical system can prove all the truths of mathematics. Gödel's second incompleteness theorem (also simplified) in turn states that no reasonable mathematical system can prove its own consistency. Another famous undecidability theorem is that the Continuum Hypothesis is neither provable nor refutable in standard set theory. Many of us logicians were first attracted to the field as students because we had heard something of these results. All research mathematicians know something of them too, and have at least a rough sense of why ‘we can't prove everything we want to prove’.

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().

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