Basic concepts in infinite set theory | Cardinal numbers | Forcing (mathematics) | Independence results | Infinity

Continuum hypothesis

In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that there is no set whose cardinality is strictly between that of the integers and the real numbers, or equivalently, that any subset of the real numbers is finite, is countably infinite, or has the same cardinality as the real numbers. In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers: , or even shorter with beth numbers: . The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900. The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in 1940. The name of the hypothesis comes from the term the continuum for the real numbers. (Wikipedia).

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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

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What is a Null Hypothesis?

Overview of null hypothesis, examples of null and alternate hypotheses, and how to write a null hypothesis statement.

From playlist Hypothesis Tests and Critical Values

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Quantum Mechanics 1.1: Introduction

In this video I provide some motivation behind the development of quantum mechanics, kicking off a new series on everything you've been wondering about quantum mechanics! Twitter: https://twitter.com/SciencePlease_

From playlist Quantum Mechanics

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What Is The Uncertainty Principle?

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From playlist Science Unplugged: Quantum Mechanics

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Albert Einstein, Holograms and Quantum Gravity

In the latest campaign to reconcile Einstein’s theory of gravity with quantum mechanics, many physicists are studying how a higher dimensional space that includes gravity arises like a hologram from a lower dimensional particle theory. Read about the second episode of the new season here:

From playlist In Theory

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Is string theory a unified theory?

Subscribe to our YouTube Channel for all the latest from World Science U. Visit our Website: http://www.worldscienceu.com/ Like us on Facebook: https://www.facebook.com/worldscienceu Follow us on Twitter: https://twitter.com/worldscienceu

From playlist Science Unplugged: String Theory

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Quantum Theory - Full Documentary HD

Check: https://youtu.be/Hs_chZSNL9I The World of Quantum - Full Documentary HD http://www.advexon.com For more Scientific DOCUMENTARIES. Subscribe for more Videos... Quantum mechanics (QM -- also known as quantum physics, or quantum theory) is a branch of physics which deals with physica

From playlist TV Appearances

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The Continuum Hypothesis

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From playlist Logic

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Colloquium MathAlp 2018 - Patrick Dehornoy

La théorie des ensembles cinquante ans après Cohen : On présentera quelques résultats de théorie des ensembles récents, avec un accent sur l'hypothèse du continu et la possibilité de résoudre la question après les résultats négatifs bien connus de Gödel et Cohen, et sur les tables de Lave

From playlist Colloquiums MathAlp

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This is part of a series of lectures on the Zermelo-Fraenkel axioms for set theory. We discuss the powerset axiom, the strongest of the ZF axioms, and explain why the notion of a powerset is so hard to pin down precisely. For the other lectures in the course see https://www.youtube.com

From playlist Zermelo Fraenkel axioms

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Real Analysis Ep 6: Countable vs uncountable

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From playlist Math 3371 (Real analysis) Fall 2020

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The Scalar Field Propagator on a Causal Set by Sumati Surya

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From playlist Bangalore Area String Meeting

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From playlist An Infinite Playlist

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Could it be that either quantum mechanics or general relativity is wrong?

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From playlist Science Unplugged: Quantum Mechanics

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A conversation between Gregory Chaitin and Stephen Wolfram, Part 2

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From playlist Conversations with Special Guests

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What is quantum mechanics? A minimal formulation (Seminar) by Pierre Hohenberg

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From playlist US-India Advanced Studies Institute: Classical and Quantum Information

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Does Infinite Cardinal Arithmetic Resemble Number Theory? - Menachem Kojman

Menachem Kojman Ben-Gurion University of the Negev; Member, School of Mathematics February 28, 2011 I will survey the development of modern infinite cardinal arithmetic, focusing mainly on S. Shelah's algebraic pcf theory, which was developed in the 1990s to provide upper bounds in infinit

From playlist Mathematics

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The Mathematical Truth | Enrico Bombieri

Enrico Bombieri, Professor Emeritus, School of Mathematics, Institute for Advanced Study http://www.ias.edu/people/faculty-and-emeriti/bombieri October 29, 2010 In this lecture, Professor Enrico Bombieri attempts to give an idea of the numerous different notions of truth in mathematics.

From playlist Mathematics

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The multiverse hypothesis: Is our universe the only one?

Support me on Patreon: https://www.patreon.com/Sabine In the past decades, the idea that our universe is only one of many, has become popular among physicists. If there are several universes, their collection is called the “multiverse”, and physicists have a few theories for this that I e

From playlist Physics

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Multiverse (set theory) | Power set | Ω-logic | Countable set | Hartogs number | Solomon Feferman | Wetzel's problem | Continuum (set theory) | Topology | Infinite set | Conjecture | Gödel's incompleteness theorems | Gödel numbering | Beth number | Mathematical analysis | Ontological maximalism | König's theorem (set theory) | Probability | Axiom of constructibility | Rational number | Martin's maximum | Forcing (mathematics) | Freiling's axiom of symmetry | Independence (mathematical logic) | Ordinal number | Inner model | Zermelo–Fraenkel set theory | Universe (mathematics) | Cardinal number | Constructible universe | Mathematics | Integer | Real number | Wacław Sierpiński | Axiom of determinacy | Aleph number | Easton's theorem | Bijection | Intuitionistic logic | Cardinality | Classical logic | Skolem's paradox | Cantor's diagonal argument | Philosophy of mathematics | Cofinality