Ordinal numbers | Mathematical notation | Proof theory
In mathematical logic and set theory, an ordinal notation is a partial function mapping the set of all finite sequences of symbols, themselves members of a finite alphabet, to a countable set of ordinals. A Gödel numbering is a function mapping the set of well-formed formulae (a finite sequence of symbols on which the ordinal notation function is defined) of some formal language to the natural numbers. This associates each well-formed formula with a unique natural number, called its Gödel number. If a Gödel numbering is fixed, then the subset relation on the ordinals induces an ordering on well-formed formulae which in turn induces a well-ordering on the subset of natural numbers. A recursive ordinal notation must satisfy the following two additional properties: 1. * the subset of natural numbers is a recursive set 2. * the induced well-ordering on the subset of natural numbers is a recursive relation There are many such schemes of ordinal notations, including schemes by Wilhelm Ackermann, Heinz Bachmann, Wilfried Buchholz, Georg Cantor, Solomon Feferman, Gerhard Jäger, Isles, Pfeiffer, Wolfram Pohlers, Kurt Schütte, Gaisi Takeuti (called ordinal diagrams), Oswald Veblen. Stephen Cole Kleene has a system of notations, called Kleene's O, which includes ordinal notations but it is not as well behaved as the other systems described here. Usually one proceeds by defining several functions from ordinals to ordinals and representing each such function by a symbol. In many systems, such as Veblen's well known system, the functions are normal functions, that is, they are strictly increasing and continuous in at least one of their arguments, and increasing in other arguments. Another desirable property for such functions is that the value of the function is greater than each of its arguments, so that an ordinal is always being described in terms of smaller ordinals. There are several such desirable properties. Unfortunately, no one system can have all of them since they contradict each other. (Wikipedia).
What is the definition of scientific notation
👉 Learn about scientific notations. Scientific notation is a convenient way of writing very large or very small numbers. A number written in scientific notation is of the form a * 10^n where a is the first non-zero number between 1 and 10, (1 included) and n is the number of digits up to t
From playlist Scientific Notation | Learn About
Determining the quotient of two numbers in scientific notation
👉 Learn how to divide numbers written in scientific notations. Scientific notation is a convenient way of writing very large or very small numbers. A number written in scientific notation is of the form a * 10^n where a is the first non-zero number between 1 and 10, (1 included) and n is t
From playlist Scientific Notation
The Meaning of Numbers – Categorical Data (1-2)
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From playlist WK1 Numbers and Variables - Online Statistics for the Flipped Classroom
Find the quotient between two numbers by converting to scientific notation
👉 Learn how to divide numbers written in scientific notations. Scientific notation is a convenient way of writing very large or very small numbers. A number written in scientific notation is of the form a * 10^n where a is the first non-zero number between 1 and 10, (1 included) and n is t
From playlist Scientific Notation
Finding the quotient for two numbers while in scientific notation
👉 Learn how to divide numbers written in scientific notations. Scientific notation is a convenient way of writing very large or very small numbers. A number written in scientific notation is of the form a * 10^n where a is the first non-zero number between 1 and 10, (1 included) and n is t
From playlist Scientific Notation
The sequence that grows remarkably large, then drops to zero!
Goodstein sequences can get larger than Graham's number and the growth rate can be faster than Ackermann’s function. In fact, these sequences grow at such an incredible rate, that the theorem literally cannot be proven using first order arithmetic and can only be proven using a stronger sy
From playlist Summer of Math Exposition 2 videos
Determine the quotient of two numbers in scientific notation
👉 Learn how to divide numbers written in scientific notations. Scientific notation is a convenient way of writing very large or very small numbers. A number written in scientific notation is of the form a * 10^n where a is the first non-zero number between 1 and 10, (1 included) and n is t
From playlist Scientific Notation
How Infinity Explains the Finite | Infinite Series
Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: https://to.pbs.org/donateinfi Peano arithmetic proves many theories in mathematics but does have its limits. In order to prove certain things you have to step beyond these axioms. Sometimes you need
From playlist An Infinite Playlist
Determine the product of two very large numbers by converting to scientific notation
👉 Learn how to multiply numbers written in scientific notations. Scientific notation is a convenient way of writing very large or very small numbers. A number written in scientific notation is of the form a * 10^n where a is the first non-zero number between 1 and 10, (1 included) and n is
From playlist Scientific Notation
Should the power class of any non-empty set even be a set? It's not in constructive Zermelo-Fraenkel, but once you add the Axiom of Choice you end up in ZFC where you have to assign it a cardinal number. But then, well-orderings on something like the reals provably exist that are not descr
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How to write a decimal in scientific notation
👉 Learn how to convert numbers to scientific notations. Scientific notation is a convenient way of writing very large or very small numbers. A number written in scientific notation is of the form a * 10^n where a is the first non-zero number between 1 and 10, (1 included) and n is the numb
From playlist Scientific Notation
Live CEOing Ep 351: Language Design in Wolfram Language [Array Processing | Part 2]
In this episode of Live CEOing, Stephen Wolfram discusses the language design of Wolfram Language functions relating to array processing and more. If you'd like to contribute to the discussion in future episodes, you can participate through this YouTube channel or through the official Twit
From playlist Behind the Scenes in Real-Life Software Design
Rick Sommer - Knuth’s Up-Arrow into the Transfinite & Beyond! - G4G14 Apr 2022
Famous for its world-record status, Graham’s number has captured the imagination of recreational mathematicians ever since being introduced by Martin Gardner in Mathematical Recreations in 1977. Knuth’s up-arrow notation builds on a nifty recursion that is used to define Graham’s number, a
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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
Learn how to write a decimal as a number in scientific notation
👉 Learn how to convert numbers to scientific notations. Scientific notation is a convenient way of writing very large or very small numbers. A number written in scientific notation is of the form a * 10^n where a is the first non-zero number between 1 and 10, (1 included) and n is the numb
From playlist Scientific Notation
A new basis theorem for ∑13 sets
Distinguished Visitor Lecture Series A new basis theorem for ∑13 sets W. Hugh Woodin Harvard University, USA and University of California, Berkeley, USA
From playlist Distinguished Visitors Lecture Series
Set Theory 1.3 : Ordinals, Supremums, and Minimums (Well Order)
In this video, I prove that a class of ordinals has a minimum and a supremum (as long as it has an upperbound). Email : fematikaqna@gmail.com Subreddit : https://www.reddit.com/r/fematika Code : https://github.com/Fematika/Animations Notes : https://docdro.id/zf39IqL
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Finding product of two numbers when they are in scientific notation
👉 Learn how to multiply numbers written in scientific notations. Scientific notation is a convenient way of writing very large or very small numbers. A number written in scientific notation is of the form a * 10^n where a is the first non-zero number between 1 and 10, (1 included) and n is
From playlist Scientific Notation
Defining Infinity | Infinite Series
Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: https://to.pbs.org/donateinfi Set theory is supposed to be a foundation of all of mathematics. How does it handle infinity? Learn through active problem-solving at Brilliant: https://brilliant.org/I
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