Large cardinals

Kunen's inconsistency theorem

In set theory, a branch of mathematics, Kunen's inconsistency theorem, proved by Kenneth Kunen, shows that several plausible large cardinal axioms are inconsistent with the axiom of choice. Some consequences of Kunen's theorem (or its proof) are: * There is no non-trivial elementary embedding of the universe V into itself. In other words, there is no Reinhardt cardinal. * If j is an elementary embedding of the universe V into an inner model M, and λ is the smallest fixed point of j above the critical point κ of j, then M does not contain the set j "λ (the image of j restricted to λ). * There is no ω-huge cardinal. * There is no non-trivial elementary embedding of Vλ+2 into itself. It is not known if Kunen's theorem still holds in ZF (ZFC without the axiom of choice), though showed that there is no definable elementary embedding from V into V. That is there is no formula J in the language of set theory such that for some parameter p∈V for all sets x∈V and y∈V: Kunen used Morse–Kelley set theory in his proof. If the proof is re-written to use ZFC, then one must add the assumption that replacement holds for formulas involving j. Otherwise one could not even show that j "λ exists as a set. The forbidden set j "λ is crucial to the proof. The proof first shows that it cannot be in M. The other parts of the theorem are derived from that. It is possible to have models of set theory that have elementary embeddings into themselves, at least if one assumes some mild large cardinal axioms. For example, if 0# exists then there is an elementary embedding from the constructible universe L into itself. This does not contradict Kunen's theorem because if 0# exists then L cannot be the whole universe of sets. (Wikipedia).

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Yuri Kifer: Nonconventional limit theorems in probability and dynamical systems

Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b

From playlist Probability and Statistics

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Examples of removable and non removable discontinuities to find limits

👉 Learn how to classify the discontinuity of a function. A function is said to be discontinuos if there is a gap in the graph of the function. Some discontinuities are removable while others are non-removable. There is also jump discontinuity. A discontinuity is removable when the denomin

From playlist Holes and Asymptotes of Rational Functions

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Learn how to find and classify the discontinuity of the function

👉 Learn how to classify the discontinuity of a function. A function is said to be discontinuous if there is a gap in the graph of the function. Some discontinuities are removable while others are non-removable. There is also jump discontinuity. A discontinuity is removable when the denomi

From playlist Holes and Asymptotes of Rational Functions

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In this short video I explain how the proof of Freiman's theorem for subsets of Z differs from the proof given earlier for subsets of F_p^N. The answer is not very much: the main differences are due to the fact that cyclic groups of prime order do not have lots of subgroups, so one has to

From playlist Introduction to Additive Combinatorics (Cambridge Part III course)

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👉 Learn how to find the removable and non-removable discontinuity of a function. A function is said to be discontinuous at a point when there is a gap in the graph of the function at that point. A discontinuity is said to be removable when there is a factor in the numerator which can cance

From playlist Find the Asymptotes of Rational Functions

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How to simplify the difference quotient of a function

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From playlist Evaluate the Limit (PC)

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Eddie Chumney in Finland 2012-05-05 PART 3/8

Eddie Chumney is one of the leading Torah teachers in the Hebraic Roots community of Messianic bible-believers today. We had the privilege and honor to have him as a guest in Finland in May 2012, and we organized a few meetings for him to speak in about Hebraic Roots of Christianity. I wil

From playlist Eddie Chumney @ Finland/Suomi @ 2012/5

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Fundamentals of Mathematics - Lecture 26: Well-Definedness

course page: https://www.uvm.edu/~tdupuy/logic/Math52-Fall2017.html videography - Eric Melton, UVM

From playlist Fundamentals of Mathematics

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From playlist Mathematics

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From playlist Infinity, and Beyond!

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Foundations - Seminar 8

Billy Price and Will Troiani present a series of seminars on foundations of mathematics. In this seminar Billy starts the proof of the completeness theorem. You can join this seminar from anywhere, on any device, at https://www.metauni.org. This video was filmed in Deprecation (https://

From playlist Foundations seminar

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From playlist Evaluate the Limit..........Help!

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From playlist Zermelo Fraenkel axioms

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Learn how to identify the discontinuities as removable or non removable

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From playlist Find the Asymptotes of Rational Functions

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Scalable Privacy-Friendly Client Cloud Computing

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From playlist Engineering

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Part2 Week9 2 15oct2021 (Least Squares)

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From playlist Part 2 lectures (2021 zoom)

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Gödel's First Incompleteness Theorem, Proof Sketch

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From playlist Infinity, and Beyond!

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Gödel's Second Incompleteness Theorem, Proof Sketch

In order for math to prove its own correctness, it would have to be incorrect. This result is Gödel’s second incompleteness theorem, and in this video, we provide a sketch of the proof. Created by: Cory Chang Produced by: Vivian Liu Script Editor: Justin Chen, Brandon Chen, Zachary Greenb

From playlist Infinity, and Beyond!

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15. NP-Completeness

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From playlist MIT 18.404J Theory of Computation, Fall 2020

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Convolution Theorem: Fourier Transforms

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From playlist Partial differential equations

Related pages

Rank-into-rank | Large cardinal | Set theory | Elementary equivalence | Constructible universe | Reinhardt cardinal | Morse–Kelley set theory | Zero sharp | Consistency | Critical point (set theory)