In set theory, Berkeley cardinals are certain large cardinals suggested by Hugh Woodin in a seminar at the University of California, Berkeley in about 1992. A Berkeley cardinal is a cardinal κ in a model of Zermelo–Fraenkel set theory with the property that for every transitive set M that includes κ and α < κ, there is a nontrivial elementary embedding of M into M with α < critical point < κ. Berkeley cardinals are a strictly stronger cardinal axiom than Reinhardt cardinals, implying that they are not compatible with the axiom of choice. A weakening of being a Berkeley cardinal is that for every binary relation R on Vκ, there is a nontrivial elementary embedding of (Vκ, R) into itself. This implies that we have elementary j1, j2, j3, ...j1: (Vκ, ∈) → (Vκ, ∈),j2: (Vκ, ∈, j1) → (Vκ, ∈, j1),j3: (Vκ, ∈, j1, j2) → (Vκ, ∈, j1, j2), and so on. This can be continued any finite number of times, and to the extent that the model has dependent choice, transfinitely. Thus, plausibly, this notion can be strengthened simply by asserting more dependent choice. While all these notions are incompatible with Zermelo–Fraenkel set theory (ZFC), their consequences do not appear to be false. There is no known inconsistency with ZFC in asserting that, for example:For every ordinal λ, there is a transitive model of ZF + Berkeley cardinal that is closed under λ sequences. (Wikipedia).
Spring is a season of renewal as campus comes to life and students buzz around.
From playlist Stanford Highlights
Stanford welcomes our international community
International students, scholars, and visitors are an important part of our diverse community at Stanford. Visit the Bechtel International Center: https://bechtel.stanford.edu/ Stanford's Office of Community Engagement: https://community.stanford.edu/
From playlist Stanford Highlights
Stanford students serve worldwide through Cardinal Service initiative
Stanford launches Cardinal Service, a bold university-wide initiative to elevate and expand service as a distinctive feature of a Stanford education.
From playlist Stanford Highlights
Stanford Athletics, Physical Education & Recreation: How It All Began, 1891-1892
Stanford Deputy Athletics Director Ray Purpur gave a talk on the early days of Stanford athletics, specifically, in 1891-1892.
From playlist Stanford Historical Society
Stanford Baccalaureate Highlights 2017
Stanford's 2017 Baccalaureate, a multifaith celebration that is a part of Commencement weekend, featured Marian Wright Edelman, a student reflection from Emma Coleman and spiritual music.
From playlist Stanford Highlights
Locke, Berkeley, & Empiricism: Crash Course Philosophy #6
This week we answer skeptics like Descartes with empiricism. Hank explains John Locke’s primary and secondary qualities and why George Berkeley doesn’t think that distinction works -- leaving us with literally nothing but our minds, ideas, and perceptions. -- Images and video via VideoBl
From playlist Philosophy
Support Vsauce, your brain, Alzheimer's research, and other YouTube educators by joining THE CURIOSITY BOX: a seasonal delivery of viral science toys made by Vsauce! A portion of all proceeds goes to Alzheimer's research and our Inquisitive Fellowship, a program that gives money and resour
From playlist Knowledge
Spiritual and religious life at Stanford
Stanford students cultivate communities of welcome and belonging, supporting vibrant religious organizations, and fostering religious engagement MB01NT8JREJQHDI
From playlist Stanford Highlights
Counting Woodin cardinals in HOD
Distinguished Visitor Lecture Series Counting Woodin cardinals in HOD W. Hugh Woodin Harvard University, USA and University of California, Berkeley, USA
From playlist Distinguished Visitors Lecture Series
Stanford welcomes the Class of 2026
Stanford president Marc Tessier-Lavigne joined residential staff to welcome new students to campus on Move-In Day 2022. MB01DGR93ME2ABC
From playlist Stanford Highlights
Andre Nies: Randomness connecting to set theory and to reverse mathematics
Abstract : I will discuss two recent interactions of the field called randomness via algorithmic tests. With Yokoyama and Triplett, I study the reverse mathematical strength of two results of analysis. (1) The Jordan decomposition theorem says that every function of bounded variation is th
From playlist Logic and Foundations
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
Andy Magid, University of Oklahoma
Andy Magid, University of Oklahoma Differential Brauer Monoids
From playlist Online Workshop in Memory of Ray Hoobler - April 30, 2020
Stanford: Pursuing the Next Great Discovery
Since Stanford first opened its doors to students and faculty, a core tenet of the university has been for its citizens to improve the world around them. Stanford researchers have achieved this goal repeatedly over the years, through the development of devices that made radar and MRI scans
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Advanced Quantum Mechanics Lecture 9
Originally presented by the Stanford Continuing Studies Program. Stanford University: http://www.stanford.edu/ Continuing Studies Program: http://csp.stanford.edu/ Stanford University Channel on YouTube: http://www.youtube.com/stanford
From playlist Stanford: Advanced Quantum Mechanics | CosmoLearning.org Physics
Caltech Convocation - 9/21/2014
Caltech's annual convocation ceremony offers a formal welcome and introduction to the Institute for our new undergraduate students and their parents, graduate students, and postdoctoral scholars. The program includes greetings from President Thomas F. Rosenbaum and Vice President of Studen
From playlist Research & Science
Infinite Sets and Foundations (Joel David Hamkins) | Ep. 17
Joel David Hamkins is a Professor of Logic with appointments in Philosophy and Mathematics at Oxford University. His main interest is in set theory. We discuss the field of set theory: what it can say about infinite sets and which issues are unresolved, and the relation of set theory to ph
From playlist Daniel Rubin Show, Full episodes
Advanced Quantum Mechanics Lecture 10
Originally presented by the Stanford Continuing Studies Program. Stanford University: http://www.stanford.edu/ Continuing Studies Program: http://csp.stanford.edu/ Stanford University Channel on YouTube: http://www.youtube.com/stanford
From playlist Stanford: Advanced Quantum Mechanics | CosmoLearning.org Physics
How the language you speak affects your thoughts
How can a simple nursery rhyme be so complicated to translate? If your language doesn't include words for numbers, are you still able to count? And why do speakers of a language without words for "left" and "right" always know which way is North? In this thought-provoking talk from TED-Ed
From playlist TED-Ed Weekend Student Talks