Topic

# Universal set

About: Universal set is a research topic. Over the lifetime, 714 publications have been published within this topic receiving 14736 citations.

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TL;DR: A vague set is a set of objects, each of which has a grade of membership whose value is a continuous subinterval of according to the inequality of the following type:

Abstract: A vague set is a set of objects, each of which has a grade of membership whose value is a continuous subinterval of

1,567 citations

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01 Jan 1980TL;DR: Descriptive set theory is the study of sets in separable, complete metric spaces that can be defined (or constructed), and so can be expected to have special properties not enjoyed by arbitrary pointsets.

Abstract: Descriptive Set Theory is the study of sets in separable, complete metric spaces that can be defined (or constructed), and so can be expected to have special properties not enjoyed by arbitrary pointsets. This subject was started by the French analysts at the turn of the 20th century, most prominently Lebesgue, and, initially, was concerned primarily with establishing regularity properties of Borel and Lebesgue measurable functions, and analytic, coanalytic, and projective sets. Its rapid development came to a halt in the late 1930s, primarily because it bumped against problems which were independent of classical axiomatic set theory. The field became very active again in the 1960s, with the introduction of strong set-theoretic hypotheses and methods from logic (especially recursion theory), which revolutionized it. This monograph develops Descriptive Set Theory systematically, from its classical roots to the modern 'effective' theory and the consequences of strong (especially determinacy) hypotheses. The book emphasizes the foundations of the subject, and it sets the stage for the dramatic results (established since the 1980s) relating large cardinals and determinacy or allowing applications of Descriptive Set Theory to classical mathematics. The book includes all the necessary background from (advanced) set theory, logic and recursion theory.

1,086 citations

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01 Jan 2012

TL;DR: The Antinomies of set theory are discussed in this article, where Axiomatic Foundations of Set Theory and Type-Theoretical Approaches to Set Theory are discussed.

Abstract: The Antinomies. Axiomatic Foundations of Set Theory. Type-Theoretical Approaches. Intuitionistic Conceptions of Mathematics. Metamathematical and Semantical Approaches. Bibliography. Indices.

428 citations

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TL;DR: The Chest of Drawers argument as mentioned in this paper is an extension of the pigeon-hole principle, which was first introduced by Dedekind in the early 1970s, and it can be expressed as follows: if sufficiently many objects are distributed over not too many classes, then at least one class contains many of these objects.

Abstract: Dedekind’s pigeon-hole principle, also known as the box argument or the chest of drawers argument (Schubfachprinzip) can be described, rather vaguely, as follows. If sufficiently many objects are distributed over not too many classes, then at least one class contains many of these objects. In 1930 F. P. Ramsey [12] discovered a remarkable extension of this principle which, in its simplest form, can be stated as follows. Let S be the set of all positive integers and suppose that all unordered pairs of distinct elements of S are distributed over two classes.

389 citations

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IBM

^{1}TL;DR: A full universal set of all-microwave gates on two superconducting single-frequency single-junction transmon qubits are characterized and a process map representation in the Pauli basis is introduced which is visually efficient and informative.

Abstract: We use quantum process tomography to characterize a full universal set of all-microwave gates on two superconducting single-frequency single-junction transmon qubits. All extracted gate fidelities, including those for Clifford group generators, single-qubit $\ensuremath{\pi}/4$ and $\ensuremath{\pi}/8$ rotations, and a two-qubit controlled-not, exceed $95%$ ($98%$), without (with) subtracting state preparation and measurement errors. Furthermore, we introduce a process map representation in the Pauli basis which is visually efficient and informative. This high-fidelity gate set serves as a critical building block towards scalable architectures of superconducting qubits for error correction schemes and pushes up on the known limits of quantum gate characterization.

345 citations