Mathematical logic | Set theory | Formal methods
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of naive set theory. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox) various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set theory is commonly employed as a foundational system for the whole of mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Besides its foundational role, set theory also provides the framework to develop a mathematical theory of infinity, and has various applications in computer science (such as in the theory of relational algebra), philosophy and formal semantics. Its foundational appeal, together with its paradoxes, its implications for the concept of infinity and its multiple applications, have made set theory an area of major interest for logicians and philosophers of mathematics. Contemporary research into set theory covers a vast array of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals. (Wikipedia).
Introduction to Set Theory (Discrete Mathematics)
Introduction to Set Theory (Discrete Mathematics) This is a basic introduction to set theory starting from the very beginning. This is typically found near the beginning of a discrete mathematics course in college or at the beginning of other advanced mathematics courses. ***************
From playlist Set Theory
Introduction to sets || Set theory Overview - Part 2
A set is the mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other #sets. The #set with no element is the empty
From playlist Set Theory
Introduction to sets || Set theory Overview - Part 1
A set is the mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other #sets. The #set with no element is the empty
From playlist Set Theory
Set Theory 1.1 : Axioms of Set Theory
In this video, I introduce the axioms of set theory and Russel's Paradox. Email : fematikaqna@gmail.com Code : https://github.com/Fematika/Animations Notes : http://docdro.id/5ITQHUW
From playlist Set Theory
Set Theory (Part 1): Notation and Operations
Please feel free to leave comments/questions on the video and practice problems below! In this video series, we'll explore the basics of set theory. I assume no experience with set theory in the video series and anyone who's "been around town" in math should understand the videos. To make
From playlist Set Theory by Mathoma
Review of set theory -- Proofs
This lecture is on Introduction to Higher Mathematics (Proofs). For more see http://calculus123.com.
From playlist Proofs
Set Theory (Part 2): ZFC Axioms
Please feel free to leave comments/questions on the video and practice problems below! In this video, I introduce some common axioms in set theory using the Zermelo-Fraenkel w/ choice (ZFC) system. Five out of nine ZFC axioms are covered and the remaining four will be introduced in their
From playlist Set Theory by Mathoma
The perfect number of axioms | Axiomatic Set Theory, Section 1.1
In this video we introduce 6 of the axioms of ZFC set theory. My Twitter: https://twitter.com/KristapsBalodi3 Intro: (0:00) The Axiom of Existence: (2:39) The Axiom of Extensionality: (4:20) The Axiom Schema of Comprehension: (6:15) The Axiom of Pair (12:16) The Axiom of Union (15:15) T
From playlist Axiomatic Set Theory
Introduction to Sets and Set Notation
This video defines a set, special sets, and set notation.
From playlist Sets (Discrete Math)
Axioms of Constructive Set Theory Explained
In this video we're going to discuss the various axiom schemes of constructive set theories and how they relate to type theory. I cover BCST, ECST, IKP, KPI, KP, CST, CZF, IZF, Mac Lane, Z and variants equi-consistent to ETCS from category theory, and then of course ZF and ZFC. The text I
From playlist Logic
Lecture 1: Invitation to topos theory
This talk introduces the motivating question for this semester of the Curry-Howard seminar, which is how to organise mathematical knowledge using topoi. The approach sketched out in the talk is via first-order theories, their associated classifying topoi, and adjoint pairs of functors betw
From playlist Topos theory seminar
Anna Marie Bohmann: Assembly in the Algebraic K-theory of Lawvere Theories
Talk by Anna Marie Bohmann in Global Noncommutative Geometry Seminar (Americas), https://globalncgseminar.org/talks/tba-30/, on April 29, 2022.
From playlist Global Noncommutative Geometry Seminar (Americas)
Infinite Sets and Foundations (Joel David Hamkins) | Ep. 17
Joel David Hamkins is a Professor of Logic with appointments in Philosophy and Mathematics at Oxford University. His main interest is in set theory. We discuss the field of set theory: what it can say about infinite sets and which issues are unresolved, and the relation of set theory to ph
From playlist Daniel Rubin Show, Full episodes
The Big (mathematical) Bang | Axiomatic Set Theory, Section 0
The introductory video for a course on the axiomatic theory of ZFC set theory. My Twitter: https://twitter.com/KristapsBalodi3 Intro: (0:00) Russel's Paradox: (2:13)
From playlist Axiomatic Set Theory
Set Theory - What is Set Theory and what is it for? Oxford Mathematics 3rd Year Student Lecture
This is the first of four lectures from Robin Knight's 3rd Year Set Theory course. Robin writes: "Infinity baffled mathematicians, and everyone else, for thousands of years. But around 1870, Georg Cantor worked out how to study infinity in a way that made sense, and created set theory. Mo
From playlist Oxford Mathematics Student Lectures - Set Theory
Regularity and non-standard models of arithmetic #PaCE1
Follow-up video: https://youtu.be/7HKnOOvssvs Discussed text, including all links: https://gist.github.com/Nikolaj-K/101c2712dc832dec4991bf568869abc8 Curt's call: https://youtu.be/V93GQaDtv8w Timestamps: 00:00:00 Introduction 00:02:55 Wittgenstein and predicates (optional) 00:11:12 Skolems
From playlist Logic
Absolute notions in model theory - M. Dzamonja - Workshop 1 - CEB T1 2018
Mirna Dzamonja (East Anglia) / 30.01.2018 The wonderful theory of stability and ranks developed for many notions in first order model theory implies that many model theoretic constructions are absolute, since they can be expressed in terms of internal properties measurable by the existenc
From playlist 2018 - T1 - Model Theory, Combinatorics and Valued fields
This is part of a series of lectures on the Zermelo-Fraenkel axioms for set theory. We discuss the axiom of foundation, which says that the membership relation is well founded, and give some examples of the bizarre things that can happen if sets are allowed to be non-well-founded. For
From playlist Zermelo Fraenkel axioms
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
Zermelo Fraenkel Separation and replacement
This is part of a series of lectures on the Zermelo-Fraenkel axioms for set theory. We discuss the axioms of separation and replacement and some of their variations. For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj52EKVgPi-p50fRP2_SbG2oi
From playlist Zermelo Fraenkel axioms