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• Ordinal_notation abstract "In mathematical logic and set theory, an ordinal notation is a partial function from the set of all finite sequences of symbols from a finite alphabet to a countable set of ordinals, and a Gödel numbering is a function from the set of well-formed formulae (a well-formed formula is a finite sequence of symbols on which the ordinal notation function is defined) of some formal language to the natural numbers. This associates each wff 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: the subset of natural numbers is a recursive set the induced well-ordering on the subset of natural numbers is a recursive relationThere 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.".
• Ordinal_notation wikiPageExternalLink ords.ps.
• Ordinal_notation wikiPageID "6108552".
• Ordinal_notation wikiPageLength "12372".
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• Ordinal_notation wikiPageRevisionID "707829203".
• Ordinal_notation wikiPageWikiLink Ackermann_ordinal.
• Ordinal_notation wikiPageWikiLink Arity.
• Ordinal_notation wikiPageWikiLink Category:Mathematical_notation.
• Ordinal_notation wikiPageWikiLink Category:Ordinal_numbers.
• Ordinal_notation wikiPageWikiLink Category:Proof_theory.
• Ordinal_notation wikiPageWikiLink Church–Kleene_ordinal.
• Ordinal_notation wikiPageWikiLink Computable_function.
• Ordinal_notation wikiPageWikiLink Epsilon_numbers_(mathematics).
• Ordinal_notation wikiPageWikiLink Feferman–Schütte_ordinal.
• Ordinal_notation wikiPageWikiLink First_uncountable_ordinal.
• Ordinal_notation wikiPageWikiLink Gaisi_Takeuti.
• Ordinal_notation wikiPageWikiLink Georg_Cantor.
• Ordinal_notation wikiPageWikiLink Gödel_numbering.
• Ordinal_notation wikiPageWikiLink Heinz_Bachmann.
• Ordinal_notation wikiPageWikiLink Kleenes_O.
• Ordinal_notation wikiPageWikiLink Kurt_Schütte.
• Ordinal_notation wikiPageWikiLink Large_countable_ordinal.
• Ordinal_notation wikiPageWikiLink Mathematical_logic.
• Ordinal_notation wikiPageWikiLink Natural_number.
• Ordinal_notation wikiPageWikiLink Ordinal.
• Ordinal_notation wikiPageWikiLink Ordinal_analysis.
• Ordinal_notation wikiPageWikiLink Ordinal_arithmetic.
• Ordinal_notation wikiPageWikiLink Oswald_Veblen.
• Ordinal_notation wikiPageWikiLink PostScript.
• Ordinal_notation wikiPageWikiLink Range_(mathematics).
• Ordinal_notation wikiPageWikiLink Recursive_set.
• Ordinal_notation wikiPageWikiLink Recursively_enumerable_set.
• Ordinal_notation wikiPageWikiLink Set_theory.
• Ordinal_notation wikiPageWikiLink Solomon_Feferman.
• Ordinal_notation wikiPageWikiLink Stephen_Cole_Kleene.
• Ordinal_notation wikiPageWikiLink Transfinite_induction.
• Ordinal_notation wikiPageWikiLink Veblen_ordinal.
• Ordinal_notation wikiPageWikiLink Wilhelm_Ackermann.
• Ordinal_notation wikiPageWikiLinkText "Ordinal notation".
• Ordinal_notation wikiPageWikiLinkText "Ordinal notation#Feferman's θ functions".
• Ordinal_notation wikiPageWikiLinkText "notation".
• Ordinal_notation wikiPageWikiLinkText "notations".
• Ordinal_notation wikiPageWikiLinkText "ordinal diagrams".
• Ordinal_notation wikiPageWikiLinkText "ordinal notation".
• Ordinal_notation wikiPageWikiLinkText "ordinal notation#Buchholz".
• Ordinal_notation wikiPageWikiLinkText "systematically naming".
• Ordinal_notation wikiPageUsesTemplate Template:Citation.
• Ordinal_notation wikiPageUsesTemplate Template:Harv.
• Ordinal_notation wikiPageUsesTemplate Template:Harvtxt.
• Ordinal_notation wikiPageUsesTemplate Template:Main.
• Ordinal_notation subject Category:Mathematical_notation.
• Ordinal_notation subject Category:Ordinal_numbers.
• Ordinal_notation subject Category:Proof_theory.
• Ordinal_notation hypernym Function.
• Ordinal_notation type Disease.
• Ordinal_notation type Proof.
• Ordinal_notation type Redirect.
• Ordinal_notation comment "In mathematical logic and set theory, an ordinal notation is a partial function from the set of all finite sequences of symbols from a finite alphabet to a countable set of ordinals, and a Gödel numbering is a function from the set of well-formed formulae (a well-formed formula is a finite sequence of symbols on which the ordinal notation function is defined) of some formal language to the natural numbers. This associates each wff with a unique natural number, called its Gödel number.".
• Ordinal_notation label "Ordinal notation".
• Ordinal_notation sameAs Q7100790.
• Ordinal_notation sameAs m.0fqgt1.
• Ordinal_notation sameAs Q7100790.
• Ordinal_notation wasDerivedFrom Ordinal_notation?oldid=707829203.
• Ordinal_notation isPrimaryTopicOf Ordinal_notation.