In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal κ is strongly inaccessible if it is uncountable, it is not a sum of fewer than κ cardinals smaller than κ, and implies . The term "inaccessible cardinal" is ambiguous. Until about 1950, it meant "weakly inaccessible cardinal", but since then it usually means "strongly inaccessible cardinal". An uncountable cardinal is weakly inaccessible if it is a regular weak limit cardinal. It is strongly inaccessible, or just inaccessible, if it is a regular strong limit cardinal (this is equivalent to the definition given above). Some authors do not require weakly and strongly inaccessible cardinals to be uncountable (in which case is strongly inaccessible). Weakly inaccessible cardinals were introduced by , and strongly inaccessible ones by and . Every strongly inaccessible cardinal is also weakly inaccessible, as every strong limit cardinal is also a weak limit cardinal. If the generalized continuum hypothesis holds, then a cardinal is strongly inaccessible if and only if it is weakly inaccessible. (aleph-null) is a regular strong limit cardinal. Assuming the axiom of choice, every other infinite cardinal number is regular or a (weak) limit. However, only a rather large cardinal number can be both and thus weakly inaccessible. An ordinal is a weakly inaccessible cardinal if and only if it is a regular ordinal and it is a limit of regular ordinals. (Zero, one, and ω are regular ordinals, but not limits of regular ordinals.) A cardinal which is weakly inaccessible and also a strong limit cardinal is strongly inaccessible. The assumption of the existence of a strongly inaccessible cardinal is sometimes applied in the form of the assumption that one can work inside a Grothendieck universe, the two ideas being intimately connected. (Wikipedia).
What's On The Other Side Of A Black Hole? Hint: There Is No Other Side
Picture an entire star collapsed down into a gravitational singularity. An object with so much mass, compressed so tightly, that nothing, not even light itself can escape its grasp. It's no surprise these objects have captured our imagination... and yet, I have a complaint. The name "
From playlist Black Holes
Huge Hidden Galactic Structure Found In The Zone of Avoidance Behind Milky Way
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seekae - void, from the sound of trees falling on people. http://www.myspace.com/seekaemusic
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Rare Hidden Black Hole Reveals Itself By Destroying a Star, But Something Doesn't Add Up
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Matthew Foreman: Welch games to Laver ideals
Recorded during the meeting "XVI International Luminy Workshop in Set Theory" the September 16, 2021 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM's Au
From playlist Logic and Foundations
Science Bulletins: The Known Universe
The Known Universe takes viewers from the Himalayas through our atmosphere and the inky black of space to the afterglow of the Big Bang. It is a short flight through the world's most complete four-dimensional map of the Universe, the Digital Universe Atlas, which is maintained and updated
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From playlist Knowledge
If you're not sure what a black hole is, or even if you just need a quick refresher, Brian Greene explains what lies behind these cosmological powerhouses. Subscribe to our YouTube Channel for all the latest from World Science U. Visit our Website: http://www.worldscienceu.com/ Like us o
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Sandra Müller: Lower bounds for the perfect set property at weakly compact cardinals
By the Cantor-Bendixson theorem, subtrees of the binary tree on $\omega$ satisfy a dichotomy - either the tree has countably many branches or there is a perfect subtree (and in particular, the tree has continuum manybranches, regardless of the size of the continuum). We generalize this to
From playlist Logic and Foundations
Assaf Rinot: Chain conditions, unbounded colorings and the C-sequence spectrum
Recording during the meeting "15th International Luminy Workshop in Set Theory" the September 23, 2019 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM's A
From playlist Logic and Foundations
Laura Fontanella: Reflection of stationary sets and the tree property at ℵω2+1
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
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NOVA | Trapped in an Elevator | Short | Smallest Room in the House
Ever been trapped in an elevator? Have you overcome a fear of them? Share your elevator story and peruse others' here: http://tiny.cc/vz1eh "Trapped in an Elevator" premieres Tuesday, November 2, 2010 at 8PM ET/PT on PBS.
From playlist Tech + Engineering
This lecture is part of an online course on the Zermelo Fraenkel axioms of set theory. This lecture gives an overview of the axioms, describes the von Neumann hierarchy, and sketches several approaches to interpreting the axioms (Platonism, von Neumann hierarchy, multiverse, formalism, pra
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Dima Sinapova : Prikry type forcing and combinatorial properties
Abstract: We will analyze consequences of various types of Prikry forcing on combinatorial properties at singular cardinals and their successors, focusing on weak square and simultaneous stationary reflection. The motivation is how much compactness type properties can be obtained at succes
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What is inside of a black hole?
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Astronomy - Ch. 23: Black Holes (3 of 10) Diagram of an Inactive Black Hole
Visit http://ilectureonline.com for more math and science lectures! To donate: http://www.ilectureonline.com/donate https://www.patreon.com/user?u=3236071 We will learn why it is impossible for anything, including light, to escape from within the event horizon of a black hole. And we wil
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Emily Riehl: On the ∞-topos semantics of homotopy type theory: The simplicial model of...- Lecture 2
HYBRID EVENT Recorded during the meeting "Logic and Interactions" the February 22, 2022 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual M
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Foundations S2 - Seminar 2 - The geometric part
A seminar series on the foundations of mathematics, by Will Troiani and Billy Snikkers. This season the focus is on the proof of the Ax-Grothendieck theorem: an injective polynomial function from affine space (over the complex numbers) to itself is surjective. This week Will proved the the
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The Milky Way: Dark Constellations, A Black Hole & Our Galaxy
The Milky Way is an iconic feature of the night sky. In this episode, we’ll help you spot the center of the galaxy and well-known nebulae in the Milky Way. And we’ll explain how galactic archaeologists like Dreia Carrillo figure out what the Milky Way is actually made of. » Subscribe to Se
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The Mathematical Infinity - Enrico Bombieri
This lecture by [Enrico Bombieri](http://www.ias.edu/people/faculty-and-emeriti/bombieri), IBM von Neumann Professor in the School of Mathematics, explores how mathematics has arrived at its present pragmatic view of infinity and some of the counterintuitive paradoxes, as well as some of t
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