Set theory | Cardinal numbers | Theorems in the foundations of mathematics
In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set , the set of all subsets of the power set of has a strictly greater cardinality than itself. For finite sets, Cantor's theorem can be seen to be true by simple enumeration of the number of subsets. Counting the empty set as a subset, a set with elements has a total of subsets, and the theorem holds because for all non-negative integers. Much more significant is Cantor's discovery of an argument that is applicable to any set, and shows that the theorem holds for infinite sets also. As a consequence, the cardinality of the real numbers, which is the same as that of the power set of the integers, is strictly larger than the cardinality of the integers; see Cardinality of the continuum for details. The theorem is named for German mathematician Georg Cantor, who first stated and proved it at the end of the 19th century. Cantor's theorem had immediate and important consequences for the philosophy of mathematics. For instance, by iteratively taking the power set of an infinite set and applying Cantor's theorem, we obtain an endless hierarchy of infinite cardinals, each strictly larger than the one before it. Consequently, the theorem implies that there is no largest cardinal number (colloquially, "there's no largest infinity"). (Wikipedia).
Cantor's theorem, formally proven
In this video we're going to give an explicit proof of Cantor's theorem and also go a little deeper in understanding the offending set in the famous diagonal construction employed here. https://en.wikipedia.org/wiki/Cantor%27s_theorem https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argu
From playlist Logic
The Lawvere fixed point theorem
In this video we prove a version of Lawveres fixed point theorem that holds in Cartesian closed categories. It's a nice construction that specializes to results such as Cantors diagonal argument and prove the the power set of a set is classically always larger than the set itself. https:/
From playlist Logic
Cantor's Theorem - A Classic Proof [ No surjection between Power Set and Set itself ]
GET 15% OFF EVERYTHING! THIS IS EPIC! https://teespring.com/stores/papaflammy?pr=PAPAFLAMMY Help me create more free content! =) https://www.patreon.com/mathable AC Playlist: https://www.youtube.com/watch?v=jmD1CWzHjzU&list=PLN2B6ZNu6xmdvtm_DdFUaHIK_VB84hG_m Let us prove Cantor's Theore
From playlist Theory and Proofs
Wilfried Sieg: The Cantor Bernstein Theorem How many proofs
The lecture was held within the framework of the Hausdorff Trimester Program: Types, Sets and Constructions. Abstract: Dedekind’s 1887-proof of the Cantor-Bernstein Theorem is based on his chain theory, not on Cantor’s well-ordering principle. A careful analysis of the proof extracts an a
From playlist Workshop: "Proofs and Computation"
The Field With One Element and The Riemann Hypothesis (Full Video)
A crash course of Deninger's program to prove the Riemann Hypothesis using a cohomological interpretation of the Riemann Zeta Function. You can Deninger talk about this in more detail here: http://swc.math.arizona.edu/dls/ Leave some comments!
From playlist Riemann Hypothesis
Cardinality and Cantor-Schroeder-Bernstein | Nathan Dalaklis
The Cantor-Schroeder-Bernstein Theorem (or CSB), is a tool used to determine if two sets have the same cardinality. It is particularly useful when equating the cardinality of sets of infinitely many elements. Here, we go over a proof of the theorem and try to illustrate the construction at
From playlist The New CHALKboard
Theory of numbers: Congruences: Euler's theorem
This lecture is part of an online undergraduate course on the theory of numbers. We prove Euler's theorem, a generalization of Fermat's theorem to non-prime moduli, by using Lagrange's theorem and group theory. As an application of Fermat's theorem we show there are infinitely many prim
From playlist Theory of numbers
Some identities involving the Riemann-Zeta function.
After introducing the Riemann-Zeta function we derive a generating function for its values at positive even integers. This generating function is used to prove two sum identities. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist The Riemann Zeta Function
Weil conjectures 1 Introduction
This talk is the first of a series of talks on the Weil conejctures. We recall properties of the Riemann zeta function, and describe how Artin used these to motivate the definition of the zeta function of a curve over a finite field. We then describe Weil's generalization of this to varie
From playlist Algebraic geometry: extra topics
Group actions on 1-manifolds: A list of very concrete open questions – Andrés Navas – ICM2018
Dynamical Systems and Ordinary Differential Equations Invited Lecture 9.8 Group actions on 1-manifolds: A list of very concrete open questions Andrés Navas Abstract: Over the last four decades, group actions on manifolds have deserved much attention by people coming from different fields
From playlist Dynamical Systems and ODE
G. Walsh - Boundaries of Kleinian groups
We study the problem of classifying Kleinian groups via the topology of their limit sets. In particular, we are interested in one-ended convex-cocompact Kleinian groups where each piece in the JSJ decomposition is a free group, and we describe interesting examples in this situation. In ce
From playlist Ecole d'été 2016 - Analyse géométrique, géométrie des espaces métriques et topologie
Real Analysis Ep 5: Cardinality
Episode 5 of my videos for my undergraduate Real Analysis course at Fairfield University. This is a recording of a live class. This episode is cardinality of sets. Class webpage: http://cstaecker.fairfield.edu/~cstaecker/courses/2020f3371/ Chris Staecker webpage: http://faculty.fairfiel
From playlist Math 3371 (Real analysis) Fall 2020
On Zaremba's Conjecture - Alex Kontorovich
Alex Kontorovich Stony Brook University April 22, 2011 Inspired by the theory of good lattice points in numerical integration, Zaremba conjectured in 1972 that for every denominator q, there is some coprime numerator p, such that the continued fraction expansion of p/q has uniformly bounde
From playlist Mathematics
A road to the infinities: Some topics in set theory by Sujata Ghosh
PROGRAM : SUMMER SCHOOL FOR WOMEN IN MATHEMATICS AND STATISTICS ORGANIZERS : Siva Athreya and Anita Naolekar DATE : 13 May 2019 to 24 May 2019 VENUE : Ramanujan Lecture Hall, ICTS Bangalore The summer school is intended for women students studying in first year B.A/B.Sc./B.E./B.Tech.
From playlist Summer School for Women in Mathematics and Statistics 2019
Real Analysis Ep 6: Countable vs uncountable
Episode 6 of my videos for my undergraduate Real Analysis course at Fairfield University. This is a recording of a live class. This episode is about countable and uncountable sets, Cantor's theorem, and the continuum hypothesis. Class webpage: http://cstaecker.fairfield.edu/~cstaecker/c
From playlist Math 3371 (Real analysis) Fall 2020
A. Wright - Mirzakhani's work on Earthquakes (Part 1)
We will give the proof of Mirzakhani's theorem that the earthquake flow and Teichmuller unipotent flow are measurably isomorphic. We will assume some familiarity with quadratic differentials, but no familiarity with earthquakes, and the first lecture will be devoted to preliminaries. The s
From playlist Ecole d'été 2018 - Teichmüller dynamics, mapping class groups and applications
Dynamical systems, fractals and diophantine approximations – Carlos Gustavo Moreira – ICM2018
Plenary Lecture 6 Dynamical systems, fractal geometry and diophantine approximations Carlos Gustavo Moreira Abstract: We describe in this survey several results relating Fractal Geometry, Dynamical Systems and Diophantine Approximations, including a description of recent results related
From playlist Plenary Lectures
What is the Riemann Hypothesis?
This video provides a basic introduction to the Riemann Hypothesis based on the the superb book 'Prime Obsession' by John Derbyshire. Along the way I look at convergent and divergent series, Euler's famous solution to the Basel problem, and the Riemann-Zeta function. Analytic continuation
From playlist Mathematics