The following system is Mendelson's (1997, 289–293) ST type theory. ST is equivalent with Russell's ramified theory plus the Axiom of reducibility. The domain of quantification is partitioned into an ascending hierarchy of types, with all individuals assigned a type. Quantified variables range over only one type; hence the underlying logic is first-order logic. ST is "simple" (relative to the type theory of Principia Mathematica) primarily because all members of the domain and codomain of any relation must be of the same type.There is a lowest type, whose individuals have no members and are members of the second lowest type. Individuals of the lowest type correspond to the urelements of certain set theories. Each type has a next higher type, analogous to the notion of successor in Peano arithmetic. While ST is silent as to whether there is a maximal type, a transfinite number of types poses no difficulty. These facts, reminiscent of the Peano axioms, make it convenient and conventional to assign a natural number to each type, starting with 0 for the lowest type. But type theory does not require a prior definition of the naturals. The symbols peculiar to ST are primed variables and infix . In any given formula, unprimed variables all have the same type, while primed variables range over the next higher type. The atomic formulas of ST are of two forms, (identity) and . The infix symbol suggests the intended interpretation, set membership. All variables appearing in the definition of identity and in the axioms Extensionality and Comprehension, range over individuals of one of two consecutive types. Only unprimed variables (ranging over the "lower" type) can appear to the left of '', whereas to its right, only primed variables (ranging over the "higher" type) can appear. The first-order formulation of ST rules out quantifying over types. Hence each pair of consecutive types requires its own axiom of Extensionality and of Comprehension, which is possible if Extensionality and Comprehension below are taken as axiom schemata "ranging over" types. * Identity, defined by . * Extensionality. An axiom schema. . Let denote any first-order formula containing the free variable . * Comprehension. An axiom schema. .Remark. Any collection of elements of the same type may form an object of the next higher type. Comprehension is schematic with respect to as well as to types. * Infinity. There exists a nonempty binary relation over the individuals of the lowest type, that is irreflexive, transitive, and strongly connected: and with codomain contained in domain.Remark. Infinity is the only true axiom of ST and is entirely mathematical in nature. It asserts that is a strict total order, with a codomain contained in its domain. If 0 is assigned to the lowest type, the type of is 3. Infinity can be satisfied only if the (co)domain of is infinite, thus forcing the existence of an infinite set. If relations are defined in terms of ordered pairs, this axiom requires a prior definition of ordered pair; the Kuratowski definition, adapted to ST, will do. The literature does not explain why the usual axiom of infinity (there exists an inductive set) of ZFC of other set theories could not be married to ST. ST reveals how type theory can be made very similar to axiomatic set theory. Moreover, the more elaborate ontology of ST, grounded in what is now called the "iterative conception of set," makes for axiom (schemata) that are far simpler than those of conventional set theories, such as ZFC, with simpler ontologies. Set theories whose point of departure is type theory, but whose axioms, ontology, and terminology differ from the above, include New Foundations and Scott–Potter set theory. (Wikipedia).
Introduction to the C programming language. Part of a larger series teaching programming. See http://codeschool.org
From playlist The C language
There are two different types of reductionism. One is called methodological reductionism, the other one theory reductionism. Methodological reductionism is about the properties of the real world. It’s about taking things apart into smaller things and finding that the smaller things determ
From playlist Philosophy of Science
David McAllester - Dependent Type Theory from the Perspective of Mathematics, Physics, and (...)
Dependent type theory imposes a type system on Zemelo-Fraenkel set theory (ZFC). From a mathematics and physics perspective dependent type theory naturally generalizes the Bourbaki notion of structure and provides a universal notion of isomorphism and symmetry. This comes with a universal
From playlist Mikefest: A conference in honor of Michael Douglas' 60th birthday
Bas Spitters: Modal Dependent Type Theory and the Cubical Model
The lecture was held within the framework of the Hausdorff Trimester Program: Types, Sets and Constructions. Abstract: In recent years we have seen several new models of dependent type theory extended with some form of modal necessity operator, including nominal type theory, guarded and c
From playlist Workshop: "Types, Homotopy, Type theory, and Verification"
On the Setoid Model of Type Theory - Erik Palmgren
Erik Palmgren University of Stockholm October 18, 2012 For more videos, visit http://video.ias.edu
From playlist Mathematics
Homotopy type theory: working invariantly in homotopy theory -Guillaume Brunerie
Short talks by postdoctoral members Topic: Homotopy type theory: working invariantly in homotopy theory Speaker: Guillaume Brunerie Affiliation: Member, School of Mathematics Date: September 26, 2017 For more videos, please visit http://video.ias.edu
From playlist Mathematics
Set Theory (Part 2): ZFC Axioms
Please feel free to leave comments/questions on the video and practice problems below! In this video, I introduce some common axioms in set theory using the Zermelo-Fraenkel w/ choice (ZFC) system. Five out of nine ZFC axioms are covered and the remaining four will be introduced in their
From playlist Set Theory by Mathoma
The Liouville conformal field theory quantum zipper - Morris Ang
Probability Seminar Topic: The Liouville conformal field theory quantum zipper Speaker: Morris Ang Affiliation: Columbia University Date: February 17, 2023 Sheffield showed that conformally welding a γ-Liouville quantum gravity (LQG) surface to itself gives a Schramm-Loewner evolution (
From playlist Mathematics
The affine Hecke category is a monoidal colimit - James Tao
Geometric and Modular Representation Theory Seminar Topic: The affine Hecke category is a monoidal colimit Speaker: James Tao Affiliation: Massachusetts Institute of Technology Date: February 24, 2021 For more video please visit http://video.ias.edu
From playlist Seminar on Geometric and Modular Representation Theory
This video describes important properties of the Laplace transform and gives some examples. @eigensteve on Twitter Brunton Website: eigensteve.com Book Website: http://databookuw.com Book PDF: http://databookuw.com/databook.pdf
From playlist Data-Driven Science and Engineering
(Part 2) The Palamite Controversy: A Thomistic Analysis by Fr. Peter Totleben, O.P.
A reading of chapter 2 (God and His Activity in the World: A Thomistic Approach) of "The Palamite Controversy: A Thomistic Analysis" by Peter Totleben, O.P. https://www.academia.edu/35580908/The_Palamite_Controversy_A_Thomistic_Analysis
From playlist Palamas and Thomism
Monadic Parsers at the Input Boundary
When reading a byte stream over the process I/O boundary, the first thing which everyone should do is to parse the byte stream with a monadic parser. The talk will discuss Processes and input byte streams. Monadic parsers. What they are and why they matter. The design and use of the pure
From playlist Functional Programming
Why Some Volcanoes Erupt And Others Don't
Episode 2 of 5 Check us out on iTunes! http://testtube.com/podcast Please Subscribe! http://testu.be/1FjtHn5 Have you ever wondered why some volcanoes erupt viciously and shoot hot ash and lava miles into the air while some just slowly ooze out of the top and sides? + + + + + +
From playlist How Volcanoes Have Shaped The Earth And Universe
Yongnam Lee: Q-Gorenstein Deformations and their applications
In this talk we will discuss Q-Gorenstein schemes and Q-Gorenstein morphisms in a general setting. Based on the notion of Q-Gorenstein morphism, we define the notion of Q-Gorenstein deformations and discuss their properties. Versal properties of Q-Gorenstein deformations and their applicat
From playlist HIM Lectures: Junior Trimester Program "Algebraic Geometry"
The Computer Chronicles - Atari ST (1989)
Special thanks to archive.org for hosting these episodes. Downloads of all these episodes and more can be found at: http://archive.org/details/computerchronicles
From playlist The Computer Chronicles 1989 Episodes
DEFCON 20: Attacking TPM Part 2: A Look at the ST19WP18 TPM Device
Speaker: CHRIS TARNOVSKY FLYLOGIC, INC. The STMicroelectronics ST19WL18P TPM die-level analysis. Companies like Atmel, Infineon and ST are pushing motherboard manufacturers to use these devices. End-users trust these devices to hold passwords and other secrets. Once more, I will show you
From playlist DEFCON 20
Yang Liu: Fully Dynamic Electrical Flows: Sparse Maxflow Faster than Goldberg-Rao
We give an algorithm for computing exact maximum flows on graphs with m edges and integer capacities in the range [1,U] in ̃O(m^((3/2) −(1/328)) log U) time. For sparse graphs with polynomially bounded integer capacities, this is the first improvement over the ̃O(m^(1.5) log U) time bou
From playlist Workshop: Continuous approaches to discrete optimization
From playlist Week 4 2015 Shorts