An Algorithmic Approach to Solve Continuum Hypothesis

The continuum hypothesis has been unsolved for hundreds of years. In other words, can I answer it completely? By refuting the culturally responsible continuum [1], one can link the problem to the mathematical continuum, and it is possible to disproof the continuum hypothesis [2] . To go ahead a step, one may extend our mathematical system (by employing a more powerful set theory) and solve the continuum problem by three conditional cases. This event is sim-ilar to the status cases in the discriminant of solving a quadratic equation. Hence, my proposed al-gorithmic flowchart can best settle and depict the problem. From the above, one can further con-clude that when people extend mathematics (like set theory — ZFC) into new systems (such as Force Axioms), experts can solve important mathematical problems (CH). Indeed, there are differ-ent types of such mathematical systems, similar to ancient mathematical notation. Hence, different cultures have different ways of representation, which is similar to a Chinese saying: “different vil-lages have different laws.” However, the primary purpose of mathematical notation was initially to remember and communicate. This event indicates that the basic purpose of developing any new mathematical system is to help solve a natural phenomenon in our universe.

Letter Games: a metamathematical taster

The Mathematical Gazette ◽

10.1017/mag.2016.109 ◽

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’.

Set theoretic naturalism

10.2307/2275672 ◽

1996 ◽

Vol 61 (2) ◽

pp. 490-514 ◽

Cited By ~ 15

Author(s):

Penelope Maddy

Keyword(s):

Set Theory ◽

Classical Logic ◽

Continuum Hypothesis ◽

Philosophical Analysis ◽

Central Question ◽

The Status ◽

Classical Mathematics ◽

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?

Kreisel, the continuum hypothesis and second order set theory

Journal of Philosophical Logic ◽

10.1007/bf00248732 ◽

1976 ◽

Vol 5 (2) ◽

Cited By ~ 19

Author(s):

Thomas Weston

Keyword(s):

Set Theory ◽

Continuum Hypothesis ◽

Second Order ◽

The Continuum ◽

Order Set

Set theoretic properties of Loeb measure

10.2307/2274471 ◽

1990 ◽

Vol 55 (3) ◽

pp. 1022-1036 ◽

Cited By ~ 2

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 ◽

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.

Forcing for the impredicative theory of classes

10.2307/2272540 ◽

1972 ◽

Vol 37 (1) ◽

pp. 1-18 ◽

Cited By ~ 9

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 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 .

Power-like models of set theory

10.2307/2694973 ◽

2001 ◽

Vol 66 (4) ◽

pp. 1766-1782 ◽

Cited By ~ 3

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 ◽

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().

The Roles of Set Theories in Mathematics

10.1093/oso/9780198748991.003.0001 ◽

2018 ◽

Author(s):

Colin McLarty

Keyword(s):

Set Theory ◽

Gauge Theories ◽

Elementary Theory ◽

Continuum Hypothesis ◽

Large Cardinals ◽

What mathematicians know and use about sets varies across branches of mathematics but rarely includes such fundamental aspects of Zermelo–Fraenkel (ZF) set theory as the iterative hierarchy. All mathematicians know and use the axioms of the Elementary Theory of the Category of Sets (ETCS), though few know ETCS or any set theory by name. The chapter depicts the iterative hierarchy of ZF and constructibility as gauge theories. Since gauge theories are prominently used in physics, so these are used in work on the continuum hypothesis, large cardinals, and provability in arithmetic. But mathematicians outside logic avoid these gauges and work with structures only up to isomorphism, as does ETCS.

Gödel, Kurt (1906–78)

Routledge Encyclopedia of Philosophy ◽

10.4324/9780415249126-y084-1 ◽

2018 ◽

Author(s):

John W. Dawson

Keyword(s):

Twentieth Century ◽

Set Theory ◽

Continuum Hypothesis ◽

Order Logic ◽

First Order Logic ◽

First Order ◽

Mathematical Platonism ◽

The Axiom Of Choice ◽

Fundamental Theorems ◽

The greatest logician of the twentieth century, Gödel is renowned for his advocacy of mathematical Platonism and for three fundamental theorems in logic: the completeness of first-order logic; the incompleteness of formalized arithmetic; and the consistency of the axiom of choice and the continuum hypothesis with the axioms of Zermelo–Fraenkel set theory.

Set Theory and C*-Algebras

Bulletin of Symbolic Logic ◽

10.2178/bsl/1174668215 ◽

2007 ◽

Vol 13 (1) ◽

pp. 1-20 ◽

Cited By ~ 10

Author(s):

Nik Weaver

Keyword(s):

Set Theory ◽

Continuum Hypothesis ◽

C Algebra ◽

Calkin Algebra ◽

C Algebras ◽

Basic Object ◽

AbstractWe survey the use of extra-set-theoretic hypotheses, mainly the continuum hypothesis, in the C*-algebra literature. The Calkin algebra emerges as a basic object of interest.

INDEFINITENESS IN SEMI-INTUITIONISTIC SET THEORIES: ON A CONJECTURE OF FEFERMAN

10.1017/jsl.2015.55 ◽

2016 ◽

Vol 81 (2) ◽

pp. 742-754 ◽

Cited By ~ 5

Author(s):

MICHAEL RATHJEN

Keyword(s):

Set Theory ◽

Continuum Hypothesis ◽

Intuitionistic Set Theory ◽

AbstractThe paper proves a conjecture of Solomon Feferman concerning the indefiniteness of the continuum hypothesis relative to a semi-intuitionistic set theory.