Logical Foundations of Mathematics and Computational Complexity: A Gentle Introduction

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The intention of having these problems collected in the book is to stimulate the creativity and the discovery of new and original methods for proving known results and establishing new ones. The book is divided into three parts, each of them containing a chapter dealing with a particular type of problems. The first chapter contains problems on limits of special sequences and Riemann integrals; the second chapter deals with the calculation of special classes of integrals involving a fractional part term; and the third chapter hosts a collection of problems on the calculation of series single or multiple involving either a numerical or a functional term.

Proof of impossibility

Fractional Part Integrals. A Bouquet of Series.

Bibliography

Elements of Classical Analysis. Stolz-Cesaro Lemma.

Presents a wide range of results in logic and computational complexity Explains the topic informally and then in more detail for the advanced reader Presents the ideas behind the theoretical concepts The two main themes of this book, logic and complexity, are both essential for understanding the main problems about the foundations of mathematics. Logical Foundations of Mathematics and Computational Complexity covers a broad spectrum of results in logic and set theory that are relevant to the foundations, as well as the results in computational complexity and the interdisciplinary area of proof complexity.

The author presents his ideas on how these areas are connected, what are the most fundamental problems and how they should be approached. In particular, he argues that complexity is as important for foundations as are the more traditional concepts of computability and provability. Emphasis is on explaining the essence of concepts and the ideas of proofs, rather than presenting precise formal statements and full proofs. Each section starts with concepts and results easily explained, and gradually proceeds to more difficult ones.

The notes after each section present some formal definitions, theorems and proofs. Logical Foundations of Mathematics and Computational Complexity is aimed at graduate students of all fields of mathematics who are interested in logic, complexity and foundations.

It will also be of interest for both physicists and philosophers who are curious to learn the basics of logic and complexity theory. Series: Lecture Notes in Mathematics, Vol. Foundation Subseries , X, p.

ISBN 10: 3319001183

Four self-contained advanced courses written by leading experts in the field materials presented with new proofs Many open problems presented Pluripotential theory is a very powerful tool in geometry, complex analysis and dynamics. This CIME course focused on complex Monge-Ampere equations, applications of pluripotential theory to Kahler geometry and algebraic geometry and to holomorphic dynamics. The contributions provide an extensive description of the theory and its very recent developments, starting from basic introductory materials and concluding with open questions in current research.

This is the first text on algebraic combinatorics targeted towards undergraduates Textbook written by the most well-known algebraic combinatorist world-wide Covers topics of Walks in graphs, cubes and Radon transform, Matrix-Tree Theorem, the Sperner property, and more Written by one of the foremost experts in the field, Algebraic Combinatorics is a unique undergraduate textbook that will prepare the next generation of pure and applied mathematicians. Readers will be able to apply their newfound knowledge to mathematical, engineering, and business models.

The text is primarily intended for use in a one-semester advanced undergraduate course in algebraic combinatorics, enumerative combinatorics, or graph theory. Prerequisites include a basic knowledge of linear algebra over a field, existence of finite fields, and rudiments of group theory.

The topics in each chapter build on one another and include extensive problem sets as well as hints to selected exercises. Key topics include walks on graphs, cubes and the Radon transform, the Matrix? Tree Theorem, de Bruijn sequences, the Erd? There are also three appendices on purely enumerative aspects of combinatorics related to the chapter material: the RSK algorithm, plane partitions, and the enumeration of labeled trees.

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Stanley has received several awards including the George Polya Prize in applied combinatorics, the Guggenheim Fellowship, and the Leroy P. Steele Prize for mathematical exposition.


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This method of proof can also be interpreted slightly differently, as the method of infinite descent. One postulates that a positive integer solution exists, whether or not it is the smallest one, and one shows that based on this solution a smaller solution must exist. But by mathematical induction it follows that a still smaller solution must exist, then a yet smaller one, and so on for an infinite number of steps.

But this contradicts the fact that one cannot find smaller and smaller positive integers indefinitely; the contradiction implies that the premise that a solution exists is wrong. There are two alternative methods of proving wrong a conjecture that something is impossible: by counterexample constructive proof and by logical contradiction non-constructive proof.

The obvious way to disprove an impossibility conjecture by providing a single counterexample. For example, Euler proposed that at least n different n th powers were necessary to sum to yet another n th power. The conjecture was disproved in with a counterexample involving a count of only four different 5th powers summing to another fifth power:.


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A proof by counterexample is a constructive proof. In contrast, a non-constructive proof that something is not impossible proceeds by showing it is logically contradictory for all possible counterexamples to be invalid: At least one of the items on a list of possible counterexamples must actually be a valid counterexample to the impossibility conjecture.

For example, a conjecture that it is impossible for an irrational power raised to an irrational power to be rational was disproved by showing that one of two possible counterexamples must be a valid counterexample, without showing which one it is. It shows that the square root of 2 cannot be expressed as the ratio of two integers counting numbers.

The proof bifurcated "the numbers" into two non-overlapping collections—the rational numbers and the irrational numbers. This bifurcation was used by Cantor in his diagonal method , which in turn was used by Turing in his proof that the Entscheidungsproblem the decision problem of Hilbert is undecidable. It is unknown when, or by whom, the "theorem of Pythagoras" was discovered. The discovery can hardly have been made by Pythagoras himself, but it was certainly made in his school.

There is a famous passage in Plato 's Theaetetus in which it is stated that Teodorus Plato's teacher proved the irrationality of. Three famous questions of Greek geometry were how:. All of these are problems in Euclidean construction , and Euclidean constructions can be done only if they involve only Euclidean numbers by definition of the latter Hardy and Wright p.

Irrational numbers can be Euclidean. A good example is the irrational number the square root of 2. It is simply the length of the hypotenuse of a right triangle with legs both one unit in length, and it can be constructed with straightedge and compass. But it was proved centuries after Euclid that Euclidean numbers cannot involve any operations other than addition, subtraction, multiplication, division, and the extraction of square roots. Both trisecting the general angle and doubling the cube require taking cube roots , which are not constructible numbers by compass and straightedge.

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A proof exists to demonstrate that any Euclidean number is an algebraic number —a number that is the solution to some polynomial equation. The Gauss-Wantzel theorem showed in that constructing an equilateral n -gon is impossible for most values of n. Nagel and Newman consider the question raised by the parallel postulate to be " The question is: can the axiom that two parallel lines " It was not until work in the nineteenth century by " Gauss, Bolyai, Lobachevsky, and Riemann, that the impossibility of deducing the parallel axiom from the others was demonstrated.

This outcome was of the greatest intellectual importance. A succinct definition is found in Principia Mathematica [7] :. Richard's paradox Then X is different from all the members of E , since, whatever finite value n may have, the n -th figure in X is different from the n -th figure in the n -th of the decimals composing E , and therefore X is different from the n -th decimal.

Nevertheless we have defined X in a finite number of words [i. Thus X both is and is not a member of E.

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To quote Nagel and Newman p. Forty-six preliminary definitions, together with several important preliminary theorems, must be mastered before the main results are reached" p. In fact, Nagel and Newman required a page introduction to their exposition of the proof. But if the reader feels strong enough to tackle the paper, Martin Davis observes that "This remarkable paper is not only an intellectual landmark, but is written with a clarity and vigor that makes it a pleasure to read" Davis in Undecidable, p.

It is recommended [ by whom? For an exposition suitable for non-specialists see Beltrami p. Also see Franzen Chapter 8 pp. Davis's older treatment approaches the question from a Turing machine viewpoint. Chaitin has written a number of books about his endeavors and the subsequent philosophic and mathematical fallout from them. A string is called algorithmically random if it cannot be produced from any shorter computer program.

While most strings are random , no particular one can be proved so, except for finitely many short ones:. Beltrami observes that "Chaitin's proof is related to a paradox posed by Oxford librarian G. Berry early in the twentieth century that asks for 'the smallest positive integer that cannot be defined by an English sentence with fewer than characters.

Introduction to Computational Complexity Theory

However, the sentence within quotation marks, which is itself a definition of the alleged number is less than characters in length! The question "Does any arbitrary "Diophantine equation" have an integer solution?

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