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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:

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rdfs:label	Ordinal notation (en)
rdfs:comment	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: (en)
dcterms:subject	Mathematical notation Proof theory Ordinal numbers
Wikipage page ID	6108552 (xsd:integer)
Wikipage revision ID	1123338742 (xsd:integer)
Link from a Wikipage to another Wikipage	Mathematical notation Computable function Proof theory Mathematical logic Church–Kleene ordinal Epsilon numbers (mathematics) Gaisi Takeuti Georg Cantor Ordinal analysis Ordinal arithmetic Arity Stephen Cole Kleene Feferman–Schütte ordinal Wilhelm Ackermann Gödel numbering Heinz Bachmann Large Veblen ordinal Nonrecursive ordinal Oswald Veblen Range of a function Bachmann–Howard ordinal Ackermann ordinal Ordinal numbers Transfinite recursion PostScript Solomon Feferman Kurt Schütte Natural number Ordinal number Recursive set Recursively enumerable set Set theory Kleene's O Small Veblen ordinal First uncountable ordinal Feferman-Schutte ordinal Takeuti–Feferman–Buchholz ordinal Veblen ordinal (disambiguation) Large countable ordinals
Link from a Wikipage to an external page	https://archive.org/details/prooftheoryintro0000pohl http://www.cs.fsu.edu/~levitz/ords.ps
sameAs	Ordinal notation Ordinal notation Ordinal notation Ordinal notation
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Harv dbt:Harvtxt dbt:Main dbt:Reflist dbt:Short_description
has abstract	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. (en)
gold:hypernym	Function
prov:wasDerivedFrom	wikipedia-en:Ordinal_notation?oldid=1123338742&ns=0
page length (characters) of wiki page	15374 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Ordinal_notation
is Link from a Wikipage to another Wikipage of	Limit ordinal Notation Ordinal collapsing function Gaisi Takeuti Gentzen's consistency proof Ordinal analysis Ordinal arithmetic Computable ordinal Psi (Greek) Wilhelm Ackermann Large countable ordinal History of mathematical notation Hyperarithmetical theory Feferman's function Buchholz psi functions Kleene's O Buchholz's notation Ordinal diagram Ordinal diagrams Ordinal notations
is Wikipage redirect of	Feferman's function Buchholz's notation Ordinal diagram Ordinal diagrams Ordinal notations
is foaf:primaryTopic of	wikipedia-en:Ordinal_notation

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