Content Protection. Read Aloud. Veel samast sarjast. Computability and Randomness. Category Theory: Edition 2. Category theory is a branch of abstract algebra with incredibly diverse applications. This text and reference book is aimed not only at mathematicians, but also researchers and students of computer science, logic, linguistics, cognitive science, philosophy, and any of the other fields in which the ideas are being applied.
Containing clear definitions of the essential concepts, illuminated with numerous accessible examples, and providing full proofs of all important propositions and theorems, this book aims to make the basic ideas, theorems, and methods of category theory understandable to this broad readership. Although assuming few mathematical pre-requisites, the standard of mathematical rigour is not compromised.
The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda's lemma; adjoints; monads. An extra topic of cartesian closed categories and the lambda-calculus is also provided - a must for computer scientists, logicians and linguists!
This Second Edition contains numerous revisions to the original text, including expanding the exposition, revising and elaborating the proofs, providing additional diagrams, correcting typographical errors and, finally, adding an entirely new section on monoidal categories. Nearly a hundred new exercises have also been added, many with solutions, to make the book more useful as a course text and for self-study. Simplicity Theory. Model theory, a major branch of mathematical logic, plays a key role connecting logic and other areas of mathematics such as algebra, geometry, analysis, and combinatorics.
Simplicity theory, a subject of model theory, studies a class of mathematical structures, called simple. The class includes all stable structures vector spaces, modules, algebraically closed fields, differentially closed fields, and so on , and also important unstable structures such as the random graph, smoothly approximated structures, pseudo-finite fields, ACFA and more. Simplicity theory supplies the uniform model theoretic points of views to such structures in addition to their own mathematical analyses.
This book starts with an introduction to the fundamental notions of dividing and forking, and covers up to the hyperdefinable group configuration theorem for simple theories. It collects up-to-date knowledge on simplicity theory and it will be useful to logicians, mathematicians and graduate students working on model theory. The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics.
In order to develop the argument fully, the author presents a detailed development of higher-order logic, including a comprehensive discussion of its semantics. Professor Shapiro demonstrates the prevalence of second-order notions in mathematics is practised, and also the extent to which mathematical concepts can be formulated in second-order languages. He shows how first-order languages are insufficient to codify many concepts in contemporary mathematics, and thus that higher-order logic is needed to fully reflect current mathematics.
Throughout, the emphasis is on discussing the philosophical and historical issues associated with this subject, and the implications that they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic as might be gained from a beginning graduate course which includes the incompleteness of arithmetic and the Lowenheim-Skolem theorems. All those concerned with the foundations of mathematics will find this a thought-provoking discussion of some of the central issues in this subject. Godel's Incompleteness Theorems.
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Kurt Godel, the greatest logician of our time, startled the world of mathematics in with his Theorem of Undecidability, which showed that some statements in mathematics are inherently "undecidable. In this introductory volume, Raymond Smullyan, himself a well-known logician, guides the reader through the fascinating world of Godel's incompleteness theorems.
The level of presentation is suitable for anyone with a basic acquaintance with mathematical logic. As a clear, concise introduction to a difficult but essential subject, the book will appeal to mathematicians, philosophers, and computer scientists. More related to stochastic process. Hyperfinite Dirichlet Forms and Stochastic Processes. This monograph treats the theory of Dirichlet forms from a comprehensive point of view, using "nonstandard analysis. The discrete infinitesimal setup makes it possible to study the diffusion and the jump part using essentially the same methods.
This setting has the advantage of being independent of special topological properties of the state space and in this sense is a natural one, valid for both finite- and infinite-dimensional spaces. Foundations of Infinitesimal Stochastic Analysis. Many detailed examples, not found in the literature, are included. It begins with a brief chapter on tools from logic and infinitesimal or non-standard analysis so that the material is accessible to beginning graduate students.
Meta Math! Gregory Chaitin. His investigations shed light on what we can ultimately know about the universe and the very nature of life. In an infectious and enthusiastic narrative, Chaitin delineates the specific intellectual and intuitive steps he took toward the discovery. He takes us to the very frontiers of scientific thinking, and helps us to appreciate the art—and the sheer beauty—in the science of math.
What Is Mathematics? For more than two thousand years a familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today, unfortunately, the traditional place of mathematics in education is in grave danger. The teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but does not lead to real understanding or to greater intellectual independence.
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This new edition of Richard Courant's and Herbert Robbins's classic work seeks to address this problem. Its goal is to put the meaning back into mathematics. Written for beginners and scholars, for students and teachers, for philosophers and engineers, What is Mathematics? Covering everything from natural numbers and the number system to geometrical constructions and projective geometry, from topology and calculus to matters of principle and the Continuum Hypothesis, this fascinating survey allows readers to delve into mathematics as an organic whole rather than an empty drill in problem solving.
Thus, A and a fortiori B , is a base for randomness. Diamond classes were investigated in [ 31 ] , [ 18 ] and elsewhere. These are collections of c. Greenberg and Nies [ 18 ] showed, for example, that the strongly jump-traceable c. The covering result answers a question by Nies, by letting C be the class of sets that are not Oberwolfach random:. The K -trivial c. The following diagram summarizes the properties we have discussed:.
As mentioned above, there is a Turing incomplete random that is L R -hard. By [ 10 ] , there is a ML-random positive density point that is not a density-one point. The rightmost vertical equivalence is due to [ 5 ] as discussed above. The other nontrivial implications were obtained in [ 3 ]. Determine which further implications hold for ML-random sets.
Computability and Randomness
Is there an L R -hard Oberwolfach random set? Is it Oberwolfach randomness, or a weaker notion? How does it relate to L R -hardness? Nies was supported by the Marsden Fund. Turetsky was supported by the Marsden Fund, via a postdoctoral scholarship. Theorem 1. In more detail, they showed the equivalence, for a set Z , of the following properties: Z is random and incomplete. Theorem 3 [ 3 ]. Theorem 4 [ 10 ]. Corollary 5.
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Theorem 6 Nies [ 30 ]. Corollary 7. Question 8. Question 9. Question Subsets of Cantor space are referred to as either sets or classes. For more on algorithmic randomness, the reader can consult the books [ 31 , 12 ] , which are the most up-to-date surveys of the field.
They actually showed the result for Lebesgue density in the context of the unit interval, and then derived the weaker dyadic variant for detail see the journal version [ 4 , Remark 3. Some generalizations of a fixed-point theorem. Bienvenu and R. Kolmogorov complexity and Solovay functions.
Bienvenu, N. Greenberg, A. Nies, and D. K -triviality, Oberwolfach randomness, and differentiability. Mathematisches Forschungsinstitut Oberwolfach, Preprint Series, Bienvenu, R. Hoelzl, J. Miller, and A. Demuth, Denjoy, and Density. In preparation, The Denjoy alternative for computable functions. A theory of program size formally identical to information theory.
Information-theoretic characterizations of recursive infinite strings. Nonrecursive infinite strings with simple initial segments. Cupping with random sets. To appear in the Proc. Density, forcing, and the covering problem.
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