IDENTIDADE, INDIVIDUALIDADE E QUASE-CONJUNTOS

This chapter prepares the reader's mind for reverse mathematics. As its name suggests, reverse mathematics seeks not theorems but the right axioms to prove theorems already known. Reverse mathematics began as a technical field of mathematical logic, but its main ideas have precedents in the ancient field of geometry and the early twentieth-century field of set theory. In geometry, the parallel axiom is the right axiom to prove many theorems of Euclidean geometry, such as the Pythagorean theorem. Set theory offers a more modern example: base theory called ZF, a theorem that ZF cannot prove (the well-ordering theorem) and the “right axiom” for proving it—the axiom of choice. From these and similar examples one can guess at a base theory for analysis, and the “right axioms” for proving some of its well-known theorems.

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Ioana's Superrigidity Theorem and Orbit Equivalence Relations

ISRN Algebra ◽

10.1155/2013/387540 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8

Author(s):

Samuel Coskey

Keyword(s):

Ergodic Theory ◽

Set Theory ◽

Equivalence Relations ◽

Orbit Equivalence ◽

Borel Reducibility ◽

Free Abelian Groups ◽

Property T ◽

Expository Paper ◽

Torsion Free

We give a survey of Adrian Ioana's cocycle superrigidity theorem for profinite actions of Property (T) groups and its applications to ergodic theory and set theory in this expository paper. In addition to a statement and proof of Ioana's theorem, this paper features the following: (i) an introduction to rigidity, including a crash course in Borel cocycles and a summary of some of the best-known superrigidity theorems; (ii) some easy applications of superrigidity, both to ergodic theory (orbit equivalence) and set theory (Borel reducibility); and (iii) a streamlined proof of Simon Thomas's theorem that the classification of torsion-free abelian groups of finite rank is intractable.

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Historical Introduction

Reverse Mathematics ◽

10.23943/princeton/9780691196411.003.0001 ◽

2019 ◽

pp. 1-25

Author(s):

John Stillwell

Keyword(s):

Twentieth Century ◽

Mathematical Logic ◽

Set Theory ◽

Euclidean Geometry ◽

Reverse Mathematics ◽

Pythagorean Theorem ◽

Base Theory ◽

The Axiom Of Choice ◽

The Right ◽

Main Ideas

This chapter prepares the reader's mind for reverse mathematics. As its name suggests, reverse mathematics seeks not theorems but the right axioms to prove theorems already known. Reverse mathematics began as a technical field of mathematical logic, but its main ideas have precedents in the ancient field of geometry and the early twentieth-century field of set theory. In geometry, the parallel axiom is the right axiom to prove many theorems of Euclidean geometry, such as the Pythagorean theorem. Set theory offers a more modern example: base theory called ZF, a theorem that ZF cannot prove (the well-ordering theorem) and the “right axiom” for proving it—the axiom of choice. From these and similar examples one can guess at a base theory for analysis, and the “right axioms” for proving some of its well-known theorems.

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One hundred and two problems in mathematical logic

Journal of Symbolic Logic ◽

10.2307/2271891 ◽

1975 ◽

Vol 40 (2) ◽

pp. 113-129 ◽

Cited By ~ 106

Author(s):

Harvey Friedman

Keyword(s):

Mathematical Logic ◽

Set Theory ◽

Model Theory ◽

Proof Theory ◽

Recursion Theory ◽

Logic Model ◽

Mathematical Statement ◽

Question Mark ◽

Truth Value ◽

Expository Paper

This expository paper contains a list of 102 problems which, at the time of publication, are unsolved. These problems are distributed in four subdivisions of logic: model theory, proof theory and intuitionism, recursion theory, and set theory. They are written in the form of statements which we believe to be at least as likely as their negations. These should not be viewed as conjectures since, in some cases, we had no opinion as to which way the problem would go.In each case where we believe a problem did not originate with us, we made an effort to pinpoint a source. Often this was a difficult matter, based on subjective judgments. When we were unable to pinpoint a source, we left a question mark. No inference should be drawn concerning the beliefs of the originator of a problem as to which way it will go (lest the originator be us).The choice of these problems was based on five criteria. Firstly, we are only including problems which call for the truth value of a particular mathematical statement. A second criterion is the extent to which the concepts involved in the statements are concepts that are well known, well denned, and well understood, as well as having been extensively considered in the literature. A third criterion is the extent to which these problems have natural, simple and attractive formulations. A fourth criterion is the extent to which there is evidence that a real difficulty exists in finding a solution. Lastly and unavoidably, the extent to which these problems are connected with the author's research interests in mathematical logic.

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Statistical Inference via Convex Optimization

10.23943/princeton/9780691197296.001.0001 ◽

2019 ◽

Author(s):

John Stillwell

Keyword(s):

Twentieth Century ◽

Mathematical Logic ◽

Convex Optimization ◽

Statistical Inference ◽

Set Theory ◽

Reverse Mathematics ◽

Foundations Of Mathematics ◽

Early Twentieth Century ◽

Novel Approach ◽

Main Ideas

Reverse mathematics is a new field that seeks to find the axioms needed to prove given theorems. Reverse mathematics began as a technical field of mathematical logic, but its main ideas have precedents in the ancient field of geometry and the early twentieth-century field of set theory. This book offers a historical and representative view, emphasizing basic analysis and giving a novel approach to logic. It concludes that mathematics is an arena where theorems cannot always be proved outright, but in which all of their logical equivalents can be found. This creates the possibility of reverse mathematics, where one seeks equivalents that are suitable as axioms. By using a minimum of mathematical logic in a well-motivated way, the book will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics.

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Reverse Mathematics

10.23943/princeton/9780691196411.001.0001 ◽

2019 ◽

Author(s):

John Stillwell

Keyword(s):

Twentieth Century ◽

Mathematical Logic ◽

Set Theory ◽

Reverse Mathematics ◽

Foundations Of Mathematics ◽

Early Twentieth Century ◽

Novel Approach ◽

Main Ideas

Reverse mathematics is a new field that seeks to find the axioms needed to prove given theorems. Reverse mathematics began as a technical field of mathematical logic, but its main ideas have precedents in the ancient field of geometry and the early twentieth-century field of set theory. This book offers a historical and representative view, emphasizing basic analysis and giving a novel approach to logic. It concludes that mathematics is an arena where theorems cannot always be proved outright, but in which all of their logical equivalents can be found. This creates the possibility of reverse mathematics, where one seeks equivalents that are suitable as axioms. By using a minimum of mathematical logic in a well-motivated way, the book will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics.

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Hypothesis Testing

Statistical Inference via Convex Optimization ◽

10.23943/princeton/9780691197296.003.0002 ◽

2019 ◽

pp. 41-184

Author(s):

John Stillwell

Keyword(s):

Twentieth Century ◽

Mathematical Logic ◽

Hypothesis Testing ◽

Set Theory ◽

Reverse Mathematics ◽

Pythagorean Theorem ◽

Base Theory ◽

The Axiom Of Choice ◽

The Right ◽

Main Ideas

This chapter prepares the reader's mind for reverse mathematics. As its name suggests, reverse mathematics seeks not theorems but the right axioms to prove theorems already known. Reverse mathematics began as a technical field of mathematical logic, but its main ideas have precedents in the ancient field of geometry and the early twentieth-century field of set theory. In geometry, the parallel axiom is the right axiom to prove many theorems of Euclidean geometry, such as the Pythagorean theorem. Set theory offers a more modern example: base theory called ZF, a theorem that ZF cannot prove (the well-ordering theorem) and the “right axiom” for proving it—the axiom of choice. From these and similar examples one can guess at a base theory for analysis, and the “right axioms” for proving some of its well-known theorems.

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Sparse Recovery via ℓ1 Minimization

Statistical Inference via Convex Optimization ◽

10.23943/princeton/9780691197296.003.0001 ◽

2019 ◽

pp. 1-40

Author(s):

John Stillwell

Keyword(s):

Twentieth Century ◽

Mathematical Logic ◽

Set Theory ◽

Reverse Mathematics ◽

Sparse Recovery ◽

ℓ1 Minimization ◽

Base Theory ◽

The Axiom Of Choice ◽

The Right ◽

Main Ideas

This chapter prepares the reader's mind for reverse mathematics. As its name suggests, reverse mathematics seeks not theorems but the right axioms to prove theorems already known. Reverse mathematics began as a technical field of mathematical logic, but its main ideas have precedents in the ancient field of geometry and the early twentieth-century field of set theory. In geometry, the parallel axiom is the right axiom to prove many theorems of Euclidean geometry, such as the Pythagorean theorem. Set theory offers a more modern example: base theory called ZF, a theorem that ZF cannot prove (the well-ordering theorem) and the “right axiom” for proving it—the axiom of choice. From these and similar examples one can guess at a base theory for analysis, and the “right axioms” for proving some of its well-known theorems.

Download Full-text