Invariant descriptive set theory is a structural complexity theory for equivalence relations arising in logic and mathematics. Many significant problems in mathematics ask for satisfactory classification of mathematical objects. These classification problems can be viewed as equivalence relations, allowing the framework of invariant descriptive set theory to be applied. In the recent years invariant descriptive set theory has been greatly advanced and successfully applied to obtain an understanding of the complexity of many meaningful mathematical classification problems.
This project seeks further development of the theory and its applications. The topics investigated in this project are interdisciplinary and bring together concepts, methods, and techniques from different areas of mathematics and logic. The invariant set theory studies Polish groups, their actions, and the orbit equivalence relations induced by these actions. It can be viewed as a complexity theory for classification problems in mathematics.
Enough techniques have been developed in this funded project and in previous, related projects, which led to the solution of a twenty-year-old open problem on the existence of surjectively universal Polish groups. In addition, a majority of open problems on the topic of universality of Polish groups have been resolved. The study of 2-colorings and marker structures on countable groups, initiated by the PI and his collaborators as a tool to understand the actions of countable groups, has been developed greatly in this funded project.
It has been proven a powerful tool to give sweeping answers to many open questions in previous studies. In addition to the known connections with symbolic dynamics, this study also established new connections of the subject with forcing and graph theory. Many such properties are studied, including inaccessible cardinals , measurable cardinals , and many more. These properties typically imply the cardinal number must be very large, with the existence of a cardinal with the specified property unprovable in Zermelo-Fraenkel set theory. Determinacy refers to the fact that, under appropriate assumptions, certain two-player games of perfect information are determined from the start in the sense that one player must have a winning strategy.
The existence of these strategies has important consequences in descriptive set theory, as the assumption that a broader class of games is determined often implies that a broader class of sets will have a topological property. The axiom of determinacy AD is an important object of study; although incompatible with the axiom of choice, AD implies that all subsets of the real line are well behaved in particular, measurable and with the perfect set property.
Invariant Descriptive Set Theory
AD can be used to prove that the Wadge degrees have an elegant structure. Paul Cohen invented the method of forcing while searching for a model of ZFC in which the continuum hypothesis fails, or a model of ZF in which the axiom of choice fails. Forcing adjoins to some given model of set theory additional sets in order to create a larger model with properties determined i. For example, Cohen's construction adjoins additional subsets of the natural numbers without changing any of the cardinal numbers of the original model.
Forcing is also one of two methods for proving relative consistency by finitistic methods, the other method being Boolean-valued models. A cardinal invariant is a property of the real line measured by a cardinal number. For example, a well-studied invariant is the smallest cardinality of a collection of meagre sets of reals whose union is the entire real line. These are invariants in the sense that any two isomorphic models of set theory must give the same cardinal for each invariant.
Many cardinal invariants have been studied, and the relationships between them are often complex and related to axioms of set theory. Set-theoretic topology studies questions of general topology that are set-theoretic in nature or that require advanced methods of set theory for their solution. Many of these theorems are independent of ZFC, requiring stronger axioms for their proof.
A famous problem is the normal Moore space question , a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC. From set theory's inception, some mathematicians have objected to it as a foundation for mathematics.
The most common objection to set theory, one Kronecker voiced in set theory's earliest years, starts from the constructivist view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle. The feasibility of constructivism as a substitute foundation for mathematics was greatly increased by Errett Bishop 's influential book Foundations of Constructive Analysis.
The scope of predicatively founded mathematics, while less than that of the commonly accepted Zermelo-Fraenkel theory, is much greater than that of constructive mathematics, to the point that Solomon Feferman has said that "all of scientifically applicable analysis can be developed [using predicative methods]". Ludwig Wittgenstein condemned set theory. He wrote that "set theory is wrong", since it builds on the "nonsense" of fictitious symbolism, has "pernicious idioms", and that it is nonsensical to talk about "all numbers".
Category theorists have proposed topos theory as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as constructivism , finite set theory, and computable set theory. An active area of research is the univalent foundations and related to it homotopy type theory. Within homotopy type theory, a set may be regarded as a homotopy 0-type, with universal properties of sets arising from the inductive and recursive properties of higher inductive types.
Principles such as the axiom of choice and the law of the excluded middle can be formulated in a manner corresponding to the classical formulation in set theory or perhaps in a spectrum of distinct ways unique to type theory. Some of these principles may be proven to be a consequence of other principles. The variety of formulations of these axiomatic principles allows for a detailed analysis of the formulations required in order to derive various mathematical results. From Wikipedia, the free encyclopedia.
This article is about the branch of mathematics. For musical set theory, see Set theory music. Branch of mathematics that studies sets. Main articles: Set mathematics and Algebra of sets. Main article: von Neumann universe.
Catalog Record: Invariant descriptive set theory and the | HathiTrust Digital Library
Main article: Infinitary combinatorics. Main article: Descriptive set theory. Main article: Fuzzy set theory. Main article: Inner model theory. Main article: Large cardinal property. Main article: Determinacy. Main article: Forcing mathematics. Main article: Cardinal invariant. Main article: Set-theoretic topology. Mathematics portal.
These are the well-known antinomies of the set of all sets that do not contain themselves Russell , of the set of all transfinte ordinal numbers Burali-Forti , and the set of all finitely definable real numbers Richard. Brouwer , von Neumann called the "overall effect of their activity. With regards to the axiomatic method employed by second group composed of Zermelo , Abraham Fraenkel and Arthur Moritz Schoenflies , von Neumann worried that "We see only that the known modes of inference leading to the antinomies fail, but who knows where there are not others?
A synopsis of the history, written by van Heijenoort, can be found in the comments that precede von Neumann's English ed. The Univalent Foundations Program.
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Institute for Advanced Study. Set theory. Set mathematics. Paradoxes Problems. This part of this talk will concentrate on models for acoustic linings that absorb sound.
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We will see how the model for the last 40 years is provably wrong ill-posed , what we can do about it, and how even today we do not have a good predictive model for sound absorption by a surface in an airflow. Metal Forming is bending bits of metal into the right shape. It is how most things from coke cans to cars are made. Finite element analysis is the tool of choice for engineers, but it is not fast enough for real-time control of metal forming processes; so these processes are not controlled. With a controlled metal forming process, we could make parts we can't today; we could reduce the energy needed to make parts; we could use more recycled metal; and, possibly, we could build machines that can make more than one part.
The missing ingredient is a sufficiently accurate, sufficiently quick model of how metal forms, which is where mathematical modelling can help.
This part of this talk will look at some recent attempts by my group at mathematical modelling of some simple metal forming processes. Abstract: Homological stability is a topological property shared by many configuration spaces and groups of matrices, or, more generally, groups of symmetries. In recent years, it has turned into a powerful computational tool. The talk will give an overview of the subject. They continue to reveal an incredibly rich and varied mathematical structure.
I will give an overview of the progress in the field over the past few years. In particular, I will discuss a fascinating recent discovery: the spectral moments of the classical random matrix ensembles are hypergeometric orthogonal polynomials belonging to the Askey scheme. This unifies different strands of current research and provides powerful analytical tools to study these quantities.
Their connected components and other basic topological properties will be discussed. We will also explain how they can be assembled to produce a space which classifies certain bundles and represents an interesting cohomology theory. A number of explicit examples will be provided for orthogonal, unitary and projective unitary groups. Topology has, for example, been successfully applied to objective classification of neuron morphologies and to automatic detection of network dynamics. In this talk I will focus on the algebraic topology of brain structure and function, describing results obtained in collaboration with the Blue Brain Project on digitally reconstructed microcircuits of neurons in the rat cortex.
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