scholarly journals Independence and consistency proofs in quadratic form theory

1991 ◽  
Vol 56 (4) ◽  
pp. 1195-1211 ◽  
Author(s):  
James E. Baumgartner ◽  
Otmar Spinas
Keyword(s):  
Quadratic Form ◽  
Set Theory ◽  
Orthogonal Group ◽  
Continuum Hypothesis ◽  
Infinite Dimension ◽  
Form Theory ◽  
Countable Dimension ◽  
Straight Lines ◽  
Consistency Proofs ◽  
The Continuum

We consider the following properties of uncountable-dimensional quadratic spaces (E, Φ):(*) For all subspaces U ⊆ E of infinite dimension: dim U˔ < dim E.(**) For all subspaces U ⊆ E of infinite dimension: dim U˔ < ℵ0.Spaces of countable dimension are the orthogonal sum of straight lines and planes, so they cannot have (*), but (**) is trivially satisfied.These properties have been considered first in [G/O] in the process of investigating the orthogonal group of quadratic spaces. It has been shown there (in ZFC) that over arbitrary uncountable fields (**)-spaces of uncountable dimension exist.In [B/G], (**)-spaces of dimension ℵ1 (so (*) = (**)) have been constructed over arbitrary finite or countable fields. But this could be done only under the assumption that the continuum hypothesis (CH) holds in the underlying set theory.

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.

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().

Fred Appenzeller. An independence result in quadratic form theory: infinitary combinatorics applied to ε-Hermitian spaces. The journal of symbolic logic, vol. 54 (1989), pp. 689–699. - Otmar Spinas. Linear topologies on sesquilinear spaces of uncountable dimension. Fundamenta mathematicae, vol. 139 (1991), pp. 119–132. - James E. Baumgartner, Matthew Foreman, and Otmar Spinas. The spectrum of the Γ-invariant of a bilinear space. Journal of algebra, vol. 189 (1997), pp. 406–418. - James E. Baumgartner and Otmar Spinas. Independence and consistency proofs in quadratic form theory. The journal of symbolic logic, vol. 56 (1991), pp. 1195–1211. - Otmar Spinas. Iterated forcing in quadratic form theory. Israel journal of mathematics, vol. 79 (1992), pp. 297–315. - Otmar Spinas. Cardinal invariants and quadratic forms. Set theory of the reals, edited by Haim Judah, Israel mathematical conference proceedings, vol. 6, Gelbart Research Institute for Mathematical Sciences, Bar-Ilan University, Ramat-Gan 1993, distributed by the American Mathematical Society, Providence, pp. 563–581. - Saharon Shelah and Otmar Spinas. Gross spaces. Transactions of the American Mathematical Society, vol. 348 (1996), pp. 4257–4277.

2001 ◽  
Vol 7 (2) ◽  
pp. 285-286
Author(s):  
Paul C. Eklof
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
American Mathematical Society ◽  
Mathematical Society ◽  
Symbolic Logic ◽  
Iterated Forcing ◽  
Independence Result ◽  
Form Theory ◽  
Mathematical Sciences ◽  
Consistency Proofs

The Roles of Set Theories in Mathematics

2018 ◽  
Author(s):  
Colin McLarty
Keyword(s):  
Set Theory ◽  
Gauge Theories ◽  
Elementary Theory ◽  
Continuum Hypothesis ◽  
Large Cardinals ◽  
The Continuum

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)

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 Continuum

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.

What did Gödel Believe and When did He believe It?

2005 ◽  
Vol 11 (2) ◽  
pp. 194-206 ◽  
Author(s):  
Martin Davis
Keyword(s):  
Continuum Hypothesis ◽  
Belief System ◽  
Completeness Theorem ◽  
Predicate Calculus ◽  
Consistency Proof ◽  
Similar Vein ◽  
Consistency Proofs ◽  
The Continuum ◽  
Detailed Questionnaire ◽  
Mathematical Systems

Gödel has emphasized the important role that his philosophical views had played in his discoveries. Thus, in a letter to Hao Wang of December 7, 1967, explaining why Skolem and others had not obtained the completeness theorem for predicate calculus, Gödel wrote:This blindness (or prejudice, or whatever you may call it) of logicians is indeed surprising. But I think the explanation is not hard to find. It lies in a widespread lack, at that time, of the required epistemological attitude toward metamathematics and toward non-finitary reasoning. …I may add that my objectivist conception of mathematics and metamathematics in general, and of transfinite reasoning in particular, was fundamental also to my other work in logic.How indeed could one think of expressing metamathematics in the mathematical systems themselves, if the latter are considered to consist of meaningless symbols which acquire some substitute of meaning only through metamathematics?Or how could one give a consistency proof for the continuum hypothesis by means of my transfinite model Δ if consistency proofs have to be finitary?In a similar vein, Gödel has maintained that the “realist” or “Platonist” position regarding sets and the transfinite with which he is identified was part of his belief system from his student days. This can be seen in Gödel's replies to the detailed questionnaire prepared by Burke Grandjean in 1974. Gödel prepared three tentative mutually consistent replies, but sent none of them.

scholarly journals Set Theory and C*-Algebras

2007 ◽  
Vol 13 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Nik Weaver
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
C Algebra ◽  
Calkin Algebra ◽  
C Algebras ◽  
Basic Object ◽  
The Continuum

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.

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