In mathematics, a subcompact cardinal is a certain kind of large cardinal number. A cardinal number κ is subcompact if and only if for every A ⊂ H(κ+) there is a non-trivial elementary embedding j:(H(μ+), B) → (H(κ+), A) (where H(κ+) is the set of all sets of cardinality hereditarily less than κ+) with critical point μ and j(μ) = κ. Analogously, κ is a quasicompact cardinal if and only if for every A ⊂ H(κ+) there is a non-trivial elementary embedding j:(H(κ+), A) → (H(μ+), B) with critical point κ and j(κ) = μ. H(λ) consists of all sets whose transitive closure has cardinality less than λ. Every quasicompact cardinal is subcompact. Quasicompactness is a strengthening of subcompactness in that it projects large cardinal properties upwards. The relationship is analogous to that of extendible versus supercompact cardinals. Quasicompactness may be viewed as a strengthened or "boldface" version of 1-extendibility. Existence of subcompact cardinals implies existence of many 1-extendible cardinals, and hence many superstrong cardinals. Existence of a 2κ-supercompact cardinal κ implies existence of many quasicompact cardinals. Subcompact cardinals are noteworthy as the least large cardinals implying a failure of the square principle. If κ is subcompact, then the square principle fails at κ. Canonical inner models at the level of subcompact cardinals satisfy the square principle at all but subcompact cardinals. (Existence of such models has not yet been proved, but in any case the square principle can be forced for weaker cardinals.) Quasicompactness is one of the strongest large cardinal properties that can be witnessed by current inner models that do not use long extenders. For current inner models, the elementary embeddings included are determined by their effect on P(κ) (as computed at the stage the embedding is included), where κ is the critical point. This prevents them from witnessing even a κ+ strongly compact cardinal κ. Subcompact and quasicompact cardinals were defined by Ronald Jensen. (Wikipedia).
Introduction to the Cardinality of Sets and a Countability Proof
Introduction to Cardinality, Finite Sets, Infinite Sets, Countable Sets, and a Countability Proof - Definition of Cardinality. Two sets A, B have the same cardinality if there is a bijection between them. - Definition of finite and infinite sets. - Definition of a cardinal number. - Discu
From playlist Set Theory
BM9.2. Cardinality 2: Infinite Sets
Basic Methods: We continue the study of cardinality with infinite sets. First the class of countably infinite sets is considered, and basic results given. Then we give examples of uncountable sets using Cantor diagonalization arguments.
From playlist Math Major Basics
Books in Browsers 2012: Craig Mod, "Subcompact Publishing"
Books in Browsers 2012: Craig Mod, "Subcompact Publishing"
From playlist Books in Browsers 2012
Directory Links and Subdir Count
More videos like this online at http://www.theurbanpenguin.com We can use the link count in a long listing of a directory to easily understand how many sub-directories exist for a given directory. Each directory has two links, the directory name itself and the name . meaning this directory
From playlist Linux
U.S. Automotive History and the Chevy Vega
Facing increasing competition from economy cars from Japan and Europe, General Motors roared into the subcompact market with an innovative design based on cutting edge technology. The Chevrolet Vega was the 1971 Motor Trend Car of the Year, but problems were soon to follow. The History Guy
From playlist Automotive history
Craig Mod Interviewed at Books In Browsers 2012
Craig Mod Interviewed at Books In Browsers 2012
From playlist Books in Browsers 2012
Math 131 Fall 2018 092118 Cardinality
Recall definitions: injective, surjective, bijective, cardinality. Definitions: finite, countable, at most countable, uncountable, sequence. Remark: a 1-1 correspondence with the natural numbers is the same thing as a bijective sequence. Theorem: Every infinite subset of a countable set
From playlist Course 7: (Rudin's) Principles of Mathematical Analysis (Fall 2018)
This video explains what a mathematical function is and how it defines a relationship between two sets, the domain and the range. It also introduces three important categories of function: injective, surjective and bijective.
From playlist Foundational Math
Lecture 25 - Computational Geometry
This is Lecture 25 of the COMP300E (Programming Challenges) course taught by Professor Steven Skiena [http://www.cs.sunysb.edu/~skiena/] at Hong Kong University of Science and Technology in 2009. The lecture slides are available at: http://www.algorithm.cs.sunysb.edu/programmingchallenges
From playlist COMP300E - Programming Challenges - 2009 HKUST
Exploring Cosmic Origins with NASA’s SPHEREx
About the size of a subcompact car, NASA’s SPHEREx space telescope will map the entire sky to study the rapid expansion of the universe after the big bang, the composition of young planetary systems, and the evolutionary history of galaxies.
From playlist Astrophysics
Clayton Christensen's Free-market Solution to Healthcare
New videos DAILY: https://bigth.ink/youtube Join Big Think Edge for exclusive videos: https://bigth.ink/Edge ---------------------------------------------------------------------------------- ---------------------------------------------------------------------------------- ABOUT BIG TH
From playlist Clayton Christensen | Big Think
What is set subtraction? In this video we go over that, the set minus set operation, and an example of subtraction in set theory. This is a handy concept to grasp to understand the complement of a set and universal sets, which I also have videos on. Links below. I hope you find this vide
From playlist Set Theory
Books in Browsers 2012: Ron Hogan, "Beatrice: My foray into self-publishing"
Books in Browsers 2012: Ron Hogan, "Beatrice: My foray into self-publishing"
From playlist Books in Browsers 2012
Definition of a Surjective Function and a Function that is NOT Surjective
We define what it means for a function to be surjective and explain the intuition behind the definition. We then do an example where we show a function is not surjective. Surjective functions are also called onto functions. Useful Math Supplies https://amzn.to/3Y5TGcv My Recording Gear ht
From playlist Injective, Surjective, and Bijective Functions
Transcendental Functions 19 The Function a to the power x.mp4
The function a to the power x.
From playlist Transcendental Functions
Why Quantum Computing Is Relevant For Developers
Will quantum computing ever become a reality, making existing software developers obsolete? This video is an excerpt from "Exploring Quantum Computing Concepts Using Java Applications" by Johan Vos. Visit http://mng.bz/698D to watch the full video. 📚📚📚 This video references the book Qua
From playlist Java
The history of how what was marketed as the “little carefree car” that was built to be as “indestructible as a Model T" became known as one of the most dangerous cars ever built deserves to be remembered. Check out our new community for fans and supporters! https://thehistoryguyguild.loc
From playlist Automotive history
Injective, Surjective and Bijective Functions (continued)
This video is the second part of an introduction to the basic concepts of functions. It looks at the different ways of representing injective, surjective and bijective functions. Along the way I describe a neat way to arrive at the graphical representation of a function.
From playlist Foundational Math
What is an Injective Function? Definition and Explanation
An explanation to help understand what it means for a function to be injective, also known as one-to-one. The definition of an injection leads us to some important properties of injective functions! Subscribe to see more new math videos! Music: OcularNebula - The Lopez
From playlist Functions
Books in Browsers 2012: Maureen Evans & Blaine Cook, "Conversation in an electronic age"
Books in Browsers 2012: Maureen Evans & Blaine Cook, "Dear Editor: conversation in an electronic age"
From playlist Books in Browsers 2012