Mathematical logic | Model theory
In mathematical logic, a theory is complete if it is consistent and for every closed formula in the theory's language, either that formula or its negation is provable. That is, for every sentence the theory contains the sentence or its negation but not both (that is, either or ). Recursively axiomatizable first-order theories that are consistent and rich enough to allow general mathematical reasoning to be formulated cannot be complete, as demonstrated by Gödel's first incompleteness theorem. This sense of complete is distinct from the notion of a complete logic, which asserts that for every theory that can be formulated in the logic, all semantically valid statements are provable theorems (for an appropriate sense of "semantically valid"). Gödel's completeness theorem is about this latter kind of completeness. Complete theories are closed under a number of conditions internally modelling the T-schema: * For a set of formulas : if and only if and , * For a set of formulas : if and only if or . Maximal consistent sets are a fundamental tool in the model theory of classical logic and modal logic. Their existence in a given case is usually a straightforward consequence of Zorn's lemma, based on the idea that a contradiction involves use of only finitely many premises. In the case of modal logics, the collection of maximal consistent sets extending a theory T (closed under the necessitation rule) can be given the structure of a model of T, called the canonical model. (Wikipedia).
Foundations of Quantum Mechanics: Completeness
Foundations of Quantum Mechanics: Completeness This lecture is a long and complex proof that every finite vector space is complete. The purpose is to demonstrate some of the methods of real and functional analysis as well as to emphasize the significance of a vector space being finite-dim
From playlist Mathematical Foundations of Quantum Mechanics
Is the search for a unified theory really a search for the Theory of Everything?
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From playlist Science Unplugged: Grand Unification
Can there ever be a final theory of physics?
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From playlist Science Unplugged: Particle Physics
Number theory Full Course [A to Z]
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure #mathematics devoted primarily to the study of the integers and integer-valued functions. Number theorists study prime numbers as well as the properties of objects made out of integers (for example, ratio
From playlist Number Theory
Why is the Empty Set a Subset of Every Set? | Set Theory, Subsets, Subset Definition
The empty set is a very cool and important part of set theory in mathematics. The empty set contains no elements and is denoted { } or with the empty set symbol ∅. As a result of the empty set having no elements is that it is a subset of every set. But why is that? We go over that in this
From playlist Set Theory
Definite Integrals with the Limit Definition - Full Tutorial
This is a complete introduction to definite integrals. It starts with the familiar formulas needed, then goes on to a formal construction of the definite integral using Riemann Sums. Several examples of finding definite integrals with the limit definition are given. If you enjoyed this vi
From playlist Math Tutorials
The big mathematics divide: between "exact" and "approximate" | Sociology and Pure Maths | NJW
Modern pure mathematics suffers from a major schism that largely goes unacknowledged: that many aspects of the subject are parading as "exact theories" when in fact they are really only "approximate theories". In this sense they can be viewed either as belonging more properly to applied ma
From playlist Sociology and Pure Mathematics
Want a career that helps solve the world's biggest problems? Then head over to https://80000hours.org/parth Over the years, many scientists have been confident that physics is almost complete, and that humanity was just a small number of discoveries away from understanding everything in t
From playlist Quantum Physics by Parth G
What is the complement of a set? Sets in mathematics are very cool, and one of my favorite thins in set theory is the complement and the universal set. In this video we will define complement in set theory, and in order to do so you will also need to know the meaning of universal set. I go
From playlist Set Theory
The abstract chromatic number - Leonardo Nagami Coregliano
Computer Science/Discrete Mathematics Seminar I Topic: The abstract chromatic number Speaker: Leonardo Nagami Coregliano Affiliation: University of Chicago Date: March 22, 2021 For more video please visit http://video.ias.edu
From playlist Mathematics
S-Matrix Theory for Massive Higher Spins and the Challenge of UV Completion by Nima Arkani Hamed
11 January 2017 to 13 January 2017 VENUE: Ramanujan Lecture Hall, ICTS, Bengaluru String theory has come a long way, from its origin in 1970's as a possible model of strong interactions, to the present day where it sheds light not only on the original problem of strong interactions, but
From playlist String Theory: Past and Present
Physics under the Gravitational Rainbow by Claudia de Rham
Program Cosmology - The Next Decade ORGANIZERS : Rishi Khatri, Subha Majumdar and Aseem Paranjape DATE : 03 January 2019 to 25 January 2019 VENUE : Ramanujan Lecture Hall, ICTS Bangalore The great observational progress in cosmology has revealed some very intriguing puzzles, the most i
From playlist Cosmology - The Next Decade
Abraham Robinson’s legacy in model theory and (...) - L. Van den Dries - Workshop 3 - CEB T1 2018
Lou Van den Dries (University of Illinois, Urbana) / 27.03.2018 Abraham Robinson’s legacy in model theory and its applications ---------------------------------- Vous pouvez nous rejoindre sur les réseaux sociaux pour suivre nos actualités. Facebook : https://www.facebook.com/InstitutHe
From playlist 2018 - T1 - Model Theory, Combinatorics and Valued fields
In this video, we give an important motivation for studying Topological Cyclic Homology, so called "trace methods". Feel free to post comments and questions at our public forum at https://www.uni-muenster.de/TopologyQA/index.php?qa=tc-lecture Homepage with further information: https://w
From playlist Topological Cyclic Homology
Foundations S2 - Seminar 3 - Skolemisation
A seminar series on the foundations of mathematics, by Will Troiani and Billy Snikkers. This season the focus is on the proof of the Ax-Grothendieck theorem: an injective polynomial function from affine space (over the complex numbers) to itself is surjective. This week Will started into t
From playlist Foundations seminar
Foundations S2 - Seminar 2 - The geometric part
A seminar series on the foundations of mathematics, by Will Troiani and Billy Snikkers. This season the focus is on the proof of the Ax-Grothendieck theorem: an injective polynomial function from affine space (over the complex numbers) to itself is surjective. This week Will proved the the
From playlist Foundations seminar
Wolfram Physics Project: Axiomatization of the Computational Universe Tuesday, Feb. 16, 2021
This is a Wolfram Physics Project working session about the axiomatization of the Computational Universe. Begins at 1:36 Originally livestreamed at: https://twitch.tv/stephen_wolfram Stay up-to-date on this project by visiting our website: http://wolfr.am/physics Check out the announceme
From playlist Wolfram Physics Project Livestream Archive
Operational K-theory - Sam Payne
Sam Payne March 13, 2015 Workshop on Chow groups, motives and derived categories More videos on http://video.ias.edu
From playlist Mathematics
Paul Arne Østvær: The motivic Hopf map and the homotopy limit problem for hermitian K theory
The lecture was held within the framework of the (Junior) Hausdorff Trimester Program Topology: "Workshop: Hermitian K-theory and trace methods" This is a report on joint work with Markus Spitzweck and Oliver Röndigs. We use the first Hopf map to solve the homotopy limit problem for K-the
From playlist HIM Lectures: Junior Trimester Program "Topology"
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From playlist Science Unplugged: Physics