Historical Introduction

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.

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 ◽

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.

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 ◽

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.

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 ◽

Early Twentieth Century ◽

Novel Approach ◽

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.

Reverse Mathematics

10.23943/princeton/9780691196411.001.0001 ◽

2019 ◽

Author(s):

John Stillwell

Keyword(s):

Twentieth Century ◽

Mathematical Logic ◽

Set Theory ◽

Reverse Mathematics ◽

Early Twentieth Century ◽

Novel Approach ◽

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.

Reverse Mathematics: The Playground of Logic

Bulletin of Symbolic Logic ◽

10.2178/bsl/1286284559 ◽

2010 ◽

Vol 16 (3) ◽

pp. 378-402 ◽

Cited By ~ 9

Author(s):

Richard A. Shore

Keyword(s):

Mathematical Logic ◽

Reverse Mathematics ◽

Computational Approach ◽

Second Order ◽

Second Order Arithmetic ◽

Base Theory ◽

Order Arithmetic

AbstractThis paper is essentially the author's Gödel Lecture at the ASL Logic Colloquium '09 in Sofia extended and supplemented by material from some other papers. After a brief description of traditional reverse mathematics, a computational approach to is presented. There are then discussions of some interactions between reverse mathematics and the major branches of mathematical logic in terms of the techniques they supply as well as theorems for analysis. The emphasis here is on ones that lie outside the usual main systems of reverse mathematics. While retaining the usual base theory and working still within second order arithmetic, theorems are described that range from those far below the usual systems to ones far above.

Kolmogorov and mathematical logic

Journal of Symbolic Logic ◽

10.2307/2275276 ◽

1992 ◽

Vol 57 (2) ◽

pp. 385-412 ◽

Cited By ~ 38

Author(s):

Vladimir A. Uspensky

Keyword(s):

Twentieth Century ◽

Mathematical Logic ◽

Graduate Education ◽

Set Theory ◽

Personal Experience ◽

Strict Sense ◽

Human Beings ◽

The Third ◽

Intellectual Power

There are human beings whose intellectual power exceeds that of ordinary men. In my life, in my personal experience, there were three such men, and one of them was Andrei Nikolaevich Kolmogorov. I was lucky enough to be his immediate pupil. He invited me to be his pupil at the third year of my being student at the Moscow University. This talk is my tribute, my homage to my great teacher.Andrei Nikolaevich Kolmogorov was born on April 25, 1903. He graduated from Moscow University in 1925, finished his post-graduate education at the same University in 1929, and since then without any interruption worked at Moscow University till his death on October 20, 1987, at the age 84½.Kolmogorov was not only one of the greatest mathematicians of the twentieth century. By the width of his scientific interests and results he reminds one of the titans of the Renaissance. Indeed, he made prominent contributions to various fields from the theory of shooting to the theory of versification, from hydrodynamics to set theory. In this talk I should like to expound his contributions to mathematical logic.Here the term “mathematical logic” is understood in a broad sense. In this sense it, like Gallia in Caesarian times, is divided into three parts:(1) mathematical logic in the strict sense, i.e. the theory of formalized languages including deduction theory,(2) the foundations of mathematics, and(3) the theory of algorithms.

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 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.

Set—Theoretical Representations of Ordered Pairs and Their Adequacy for the Logic of Relations

Canadian Journal of Philosophy ◽

10.1080/00455091.1982.10715803 ◽

1982 ◽

Vol 12 (2) ◽

pp. 353-374 ◽

Cited By ~ 5

Author(s):

Randall R. Dipert

Keyword(s):

Twentieth Century ◽

Mathematical Logic ◽

Set Theory ◽

Norbert Wiener ◽

Precise Formulation ◽

Definition Of ◽

Ontological Economy ◽

And Function ◽

Ordered Pairs

One of the most significant discoveries of early twentieth century mathematical logic was a workable definition of ‘ordered pair’ totally within set theory. Norbert Wiener, and independently Casimir Kuratowski, are usually credited with this discovery. A definition of ‘ordered pair’ held the key to the precise formulation of the notions of ‘relation’ and ‘function’ — both of which are probably indispensable for an understanding of the foundations of mathematics. The set-theoretic definition of ‘ordered pair’ thus turned out to be a key victory for logicism, providing one admits set theory is logic. The definition also was instrumental in achieving the appearance of ontological economy — since it seemed only sets were needed — although this feature was emphasized only later.

Set theory, different systems of

Routledge Encyclopedia of Philosophy ◽

10.4324/9780415249126-y025-1 ◽

2018 ◽

Author(s):

Michael Potter

Keyword(s):

Twentieth Century ◽

Mathematical Logic ◽

Set Theory ◽

The Other ◽

Von Neumann ◽

Class Theory ◽

Iterative Conception ◽

The One ◽

Limitation Of Size ◽

As If

To begin with we shall use the word ‘collection’ quite broadly to mean anything the identity of which is solely a matter of what its members are (including ‘sets’ and ‘classes’). Which collections exist? Two extreme positions are initially appealing. The first is to say that all do. Unfortunately this is inconsistent because of Russell’s paradox: the collection of all collections which are not members of themselves does not exist. The second is to say that none do, but to talk as if they did whenever such talk can be shown to be eliminable and therefore harmless. This is consistent but far too weak to be of much use. We need an intermediate theory. Various theories of collections have been proposed since the start of the twentieth century. What they share is the axiom of ‘extensionality’, which asserts that any two sets which have exactly the same elements must be identical. This is just a matter of definition: objects which do not satisfy extensionality are not collections. Beyond extensionality, theories differ. The most popular among mathematicians is Zermelo–Fraenkel set theory (ZF). A common alternative is von Neumann–Bernays–Gödel class theory (NBG), which allows for the same sets but also has proper classes, that is, collections whose members are sets but which are not themselves sets (such as the class of all sets or the class of all ordinals). Two general principles have been used to motivate the axioms of ZF and its relatives. The first is the iterative conception, according to which sets occur cumulatively in layers, each containing all the members and subsets of all previous layers. The second is the doctrine of limitation of size, according to which the ‘paradoxical sets’ (that is, the proper classes of NBG) fail to be sets because they are in some sense too big. Neither principle is altogether satisfactory as a justification for the whole of ZF: for example, the replacement schema is motivated only by limitation of size; and ‘foundation’ is motivated only by the iterative conception. Among the other systems of set theory to have been proposed, the one that has received widespread attention is Quine’s NF (from the title of an article, ‘New Foundations for Mathematical Logic’), which seeks to avoid paradox by means of a syntactic restriction but which has not been provided with an intuitive justification on the basis of any conception of set. It is known that if NF is consistent then ZF is consistent, but the converse result has still not been proved.