Epsilon number

Ordinal numbers

In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map. Consequently, they are not reachable from 0 via a finite series of applications of the chosen exponential map and of "weaker" operations like addition and multiplication. The original epsilon numbers were introduced by Georg Cantor in the context of ordinal arithmetic; they are the ordinal numbers ε that satisfy the equation in which ω is the smallest infinite ordinal. The least such ordinal is ε0 (pronounced epsilon nought or epsilon zero), which can be viewed as the "limit" obtained by transfinite recursion from a sequence of smaller limit ordinals: where sup is the supremum function, which is equivalent to set union in the case of the von Neumann representation of ordinals. Larger ordinal fixed points of the exponential map are indexed by ordinal subscripts, resulting in . The ordinal ε0 is still countable, as is any epsilon number whose index is countable (there exist uncountable ordinals, and uncountable epsilon numbers whose index is an uncountable ordinal). The smallest epsilon number ε0 appears in many induction proofs, because for many purposes, transfinite induction is only required up to ε0 (as in Gentzen's consistency proof and the proof of Goodstein's theorem). Its use by Gentzen to prove the consistency of Peano arithmetic, along with Gödel's second incompleteness theorem, show that Peano arithmetic cannot prove the well-foundedness of this ordering (it is in fact the least ordinal with this property, and as such, in proof-theoretic ordinal analysis, is used as a measure of the strength of the theory of Peano arithmetic). Many larger epsilon numbers can be defined using the Veblen function. A more general class of epsilon numbers has been identified by John Horton Conway and Donald Knuth in the surreal number system, consisting of all surreals that are fixed points of the base ω exponential map x → ωx. defined gamma numbers (see additively indecomposable ordinal) to be numbers γ>0 such that α+γ=γ whenever α<γ, and delta numbers (see multiplicatively indecomposable ordinals) to be numbers δ>1 such that αδ=δ whenever 0<α<δ, and epsilon numbers to be numbers ε>2 such that αε=ε whenever 1<α<ε. His gamma numbers are those of the form ωβ, and his delta numbers are those of the form ωωβ. (Wikipedia).

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This video introduces the formal definition for the limit of a function at a point. Presented by Norman Wildberger of the School of Mathematics and Statistics, UNSW.

From playlist Mathematics 1A (Calculus)

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From playlist Calculus

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From playlist Calculus I - Complete Course Under Construction

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From playlist Epsilon-Delta definition of limits

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From playlist Epsilon-Delta definition of limits

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This video explains how to determine which delta values satisfy a given epsilon of a limit. http://mathispower4u.com

From playlist Limits

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From playlist Limits

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From playlist Epsilon-Delta definition of limits

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From playlist Real Analysis

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From playlist Real Analysis

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From playlist MIT 18.217 Graph Theory and Additive Combinatorics, Fall 2019

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Debabrota Basu (6/17/20): Epsilon-net induced lazy witness complex for topological data analysis

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From playlist AATRN 2020

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This video explains how to determine an expression of delta for an arbitrary epsilon that can be used to prove a limit exists. http://mathispower4u.com

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From playlist Course 6: Introduction to Analysis

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From playlist Discrete Structures

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From playlist Discrete Structures

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From playlist Calculus

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Real Analysis Playlist: https://www.youtube.com/watch?v=L-XLcmHwoh0&list=PL22w63XsKjqxqaF-Q7MSyeSG1W1_xaQoS ⭐Support the channel⭐ Patreon: https://www.patreon.com/michaelpennmath Merch: https://teespring.com/stores/michael-penn-math My amazon shop: https://www.amazon.com/shop/michaelpenn

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Epsilon number

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