Misplaced Pages

#507492

29-512: (Redirected from Eo ) Not to be confused with E0 (disambiguation) (E followed by zero). [REDACTED] Look up eo  or EO in Wiktionary, the free dictionary. Eo or EO may refer to: Businesses and organizations [ edit ] Education Otherwise , a home education organization Elevorganisasjonen , a Norwegian student organization Entrepreneurs' Organization ,

58-579: A cipher used in the Bluetooth protocol E0 (robot) , a 1986 humanoid robot by Honda E , in electrochemistry, the standard electrode potential , measuring individual potential of a reversible electrode at standard state E0, the digital carrier for audio, specified in G.703 E0, Eos Airlines IATA code E0, ethanol-free gasoline, see REC-90 e 0 , in demographics, the life expectancy of an individual at birth (age zero) E00, Cretinism ICD-10 code E00, ECO code for certain variations of

87-563: A higher order than taking the supremum of an exponential series. The following facts about epsilon numbers are straightforward to prove: Any epsilon number ε has Cantor normal form ε = ω ε {\displaystyle \varepsilon =\omega ^{\varepsilon }} , which means that the Cantor normal form is not very useful for epsilon numbers. The ordinals less than ε 0 , however, can be usefully described by their Cantor normal forms, which leads to

116-526: A major Christian branch. EO (film) , a 2022 film EO (rapper) , a rapper from London Eo (river) , a river in Spain Eo (instrument) , a Korean percussion instrument East Otago , part of New Zealand's South Island Captain EO , a 3D film Equal opportunity , a term related to civil rights Equipment Operator (US Navy) , a Seabee occupational rating in the U.S. Navy Executive Officer ,

145-482: A natural generalisation of the Cantor normal form for surreal numbers. It is natural to consider any fixed point of this expanded map to be an epsilon number, whether or not it happens to be strictly an ordinal number. Some examples of non-ordinal epsilon numbers are and There is a natural way to define ε n {\displaystyle \varepsilon _{n}} for every surreal number n , and

174-478: A new root. (This has the consequence that the number 0 is represented by a single root while the number 1 = ω 0 {\displaystyle 1=\omega ^{0}} is represented by a tree containing a root and a single leaf.) An order on the set of finite rooted trees is defined recursively: we first order the subtrees joined to the root in decreasing order, and then use lexicographic order on these ordered sequences of subtrees. In this way

203-650: A nonprofit network Evangelische Omroep , a public broadcaster in the Netherlands Executive Outcomes , a South African military company Express One International , an airline Hewa Bora Airways (IATA code EO), a defunct airline in the Democratic Republic of the Congo Language [ edit ] Eo (constructed language) Esperanto (ISO code EO), a constructed language eo (digraph) , represents

232-693: A normal function, whose fixed points form a normal function; this is known as the Veblen hierarchy (the Veblen functions with base φ 0 ( α ) = ω ). In the notation of the Veblen hierarchy, the epsilon mapping is φ 1 , and its fixed points are enumerated by φ 2 . Continuing in this vein, one can define maps φ α for progressively larger ordinals α (including, by this rarefied form of transfinite recursion, limit ordinals), with progressively larger least fixed points φ α +1 (0) . The least ordinal not reachable from 0 by this procedure—i. e.,

261-705: A person responsible for the running of an organization Executive order (United States) , a directive issued by the President of the United States Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title EO . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=EO&oldid=1239149720 " Category : Disambiguation pages Hidden categories: Short description

290-555: A representation of ε 0 as the ordered set of all finite rooted trees , as follows. Any ordinal α < ε 0 {\displaystyle \alpha <\varepsilon _{0}} has Cantor normal form α = ω β 1 + ω β 2 + ⋯ + ω β k {\displaystyle \alpha =\omega ^{\beta _{1}}+\omega ^{\beta _{2}}+\cdots +\omega ^{\beta _{k}}} where k

319-563: A single or two vowels in some languages Eo (hangul) vowel in Korean Hangul Science and technology [ edit ] Computing [ edit ] EO Personal Communicator , an early tablet computer produced by AT&T subsidiary EO, Inc. Eight Ones , a character code A line of tablet computers made by TabletKiosk Other uses in science and technology [ edit ] "eo-" (derived from "Eos", meaning "dawn"), used to describe many early animals in

SECTION 10

#1732779895508

348-1339: Is ε 0 (pronounced epsilon nought (chiefly British), epsilon naught (chiefly American), 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 , 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 ε 1 , ε 2 , … , ε ω , ε ω + 1 , … , ε ε 0 , … , ε ε 1 , … , ε ε ε ⋅ ⋅ ⋅ , … ζ 0 = φ 2 ( 0 ) {\displaystyle \varepsilon _{1},\varepsilon _{2},\ldots ,\varepsilon _{\omega },\varepsilon _{\omega +1},\ldots ,\varepsilon _{\varepsilon _{0}},\ldots ,\varepsilon _{\varepsilon _{1}},\ldots ,\varepsilon _{\varepsilon _{\varepsilon _{\cdot _{\cdot _{\cdot }}}}},\ldots \zeta _{0}=\varphi _{2}(0)} . The ordinal ε 0

377-481: Is a natural number and β 1 , … , β k {\displaystyle \beta _{1},\ldots ,\beta _{k}} are ordinals with α > β 1 ≥ ⋯ ≥ β k {\displaystyle \alpha >\beta _{1}\geq \cdots \geq \beta _{k}} , uniquely determined by α {\displaystyle \alpha } . Each of

406-554: Is a fixed point not only of base ω exponentiation but also of base δ exponentiation for all ordinals 1 < δ < ε β {\displaystyle 1<\delta <\varepsilon _{\beta }} . Since the epsilon numbers are an unbounded subclass of the ordinal numbers, they are enumerated using the ordinal numbers themselves. For any ordinal number β {\displaystyle \beta } , ε β {\displaystyle \varepsilon _{\beta }}

435-462: Is different from Wikidata All article disambiguation pages All disambiguation pages E0 (disambiguation) (Redirected from E0 (disambiguation) ) E0 or E00 can refer to: ε 0 , in mathematics, the smallest member of the epsilon numbers , a type of ordinal number ε 0 , in physics, vacuum permittivity , the absolute dielectric permittivity of classical vacuum E0 (cipher) ,

464-509: Is still countable , as is any epsilon number whose index is countable. Uncountable ordinals also exist, along with uncountable epsilon numbers whose index is an uncountable ordinal. The smallest epsilon number ε 0 appears in many induction proofs, because for many purposes transfinite induction is only required up to ε 0 (as in Gentzen's consistency proof and the proof of Goodstein's theorem ). Its use by Gentzen to prove

493-520: Is the least epsilon number (fixed point of the exponential map) not already in the set { ε δ ∣ δ < β } {\displaystyle \{\varepsilon _{\delta }\mid \delta <\beta \}} . It might appear that this is the non-constructive equivalent of the constructive definition using iterated exponentiation; but the two definitions are equally non-constructive at steps indexed by limit ordinals, which represent transfinite recursion of

522-543: The Queen's Pawn Game chess opening Enemy Zero , a 1996 Japanese horror video game for the Sega Saturn See also [ edit ] 0E (disambiguation) [REDACTED] Topics referred to by the same term This disambiguation page lists articles associated with the same title formed as a letter–number combination. If an internal link led you here, you may wish to change the link to point directly to

551-699: 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 → ω . Hessenberg (1906) 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 ordinal ) to be numbers δ > 1 such that αδ = δ whenever 0 < α < δ , and epsilon numbers to be numbers ε > 2 such that α = ε whenever 1 < α < ε . His gamma numbers are those of

580-433: The fixed-point lemma for normal functions . When α = ω {\displaystyle \alpha =\omega } , these fixed points are precisely the ordinal epsilon numbers. Because a different sequence with the same supremum, ε 1 {\displaystyle \varepsilon _{1}} , is obtained by starting from 0 and exponentiating with base ε 0 instead: Generally,

609-401: 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

SECTION 20

#1732779895508

638-430: The epsilon number ε β {\displaystyle \varepsilon _{\beta }} indexed by any ordinal that has an immediate predecessor β − 1 {\displaystyle \beta -1} can be constructed similarly. In particular, whether or not the index β is a limit ordinal, ε β {\displaystyle \varepsilon _{\beta }}

667-430: The form ω , and his delta numbers are those of the form ω . The standard definition of ordinal exponentiation with base α is: From this definition, it follows that for any fixed ordinal α > 1 , the mapping β ↦ α β {\displaystyle \beta \mapsto \alpha ^{\beta }} is a normal function , so it has arbitrarily large fixed points by

696-493: The fossil record Eoarchean , the first era of the Archean Eon Earth observation Electro-optics Eosinophilic oesophagitis , or Eosinophilic esophagitis, an allergic inflammatory condition of the esophagus Ethylene oxide , a chemical compound Extremal optimization , a type of optimization heuristic inspired by self-organized criticality Other uses [ edit ] Eastern Orthodoxy ,

725-621: The image of φ β for every β ≤ Γ 0 , including of the map φ 1 that enumerates epsilon numbers. In On Numbers and Games , the classic exposition on surreal numbers , John Horton Conway provided a number of examples of concepts that had natural extensions from the ordinals to the surreals. One such function is the ω {\displaystyle \omega } -map n ↦ ω n {\displaystyle n\mapsto \omega ^{n}} ; this mapping generalises naturally to include all surreal numbers in its domain , which in turn provides

754-437: The intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=E0&oldid=1237575759 " Category : Letter–number combination disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Epsilon numbers (mathematics) in which ω is the smallest infinite ordinal. The least such ordinal

783-709: The least ordinal α for which φ α (0) = α , or equivalently the first fixed point of the map α ↦ φ α ( 0 ) {\displaystyle \alpha \mapsto \varphi _{\alpha }(0)} —is the Feferman–Schütte ordinal Γ 0 . In a set theory where such an ordinal can be proved to exist, one has a map Γ that enumerates the fixed points Γ 0 , Γ 1 , Γ 2 , ... of α ↦ φ α ( 0 ) {\displaystyle \alpha \mapsto \varphi _{\alpha }(0)} ; these are all still epsilon numbers, as they lie in

812-440: The ordinals β 1 , … , β k {\displaystyle \beta _{1},\ldots ,\beta _{k}} in turn has a similar Cantor normal form. We obtain the finite rooted tree representing α by joining the roots of the trees representing β 1 , … , β k {\displaystyle \beta _{1},\ldots ,\beta _{k}} to

841-413: The set of all finite rooted trees becomes a well-ordered set which is order isomorphic to ε 0 . This representation is related to the proof of the hydra theorem , which represents decreasing sequences of ordinals as a graph-theoretic game. The fixed points of the "epsilon mapping" x ↦ ε x {\displaystyle x\mapsto \varepsilon _{x}} form

#507492