Everything about Ordinal notation
Type of mathematical function
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 an injective function mapping the set of well-formed formulae 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:the subset of natural numbers is a recursive set the induced well-ordering on the subset of natural numbers is a recursive relation Wikipedia →
🧭 Start here: a 6-step reading path
- Read the overview: Ordinal notationA 5-minute orientation
- Pick up a classic book: Bezeichnungssysteme für OrdinalzahlenThe most-reprinted book on the subject
- Read the landmark paper: On notation for ordinal numbersCited 470 times
- Go sideways: Nonrecursive ordinalA closely related topic
- Go sideways: Small Veblen ordinalA closely related topic
📖 Definitions
No dictionary entry for this exact phrase.
🏆 Landmark papers (most cited)
On notation for ordinal numbers
On elementary theories of ordinal notation systems based on reflection principles
On Wainer's notation for a minimal subrecursive inaccessible ordinal
Ordinal notation systems corresponding to Friedman’s linearized well-partial-orders with gap-condition
Limitwise monotonic functions relative to the Kleene’s Ordinal Notation System
Generalized ordinal notation.
Constructive Ordinal Notation Systems
NOTATIONAL SYMBOLS IN CLASSIFICATION PART V: SIGNPOSTED AND RETROACTIVE NOTATION, AND PART VI: PRONOUNCEABLE RETROACTIVE ORDINAL NOTATION
📚 Books worth reading
- 📕Bezeichnungssysteme für OrdinalzahlenHelmuth Pfeiffer · 1973
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
🔗 Related topics
- Nonrecursive ordinal In mathematics, particularly set theory, non-recursive ordinals are large countable ordinals greater than all the recursive ordinals, and therefore can…
- Small Veblen ordinal In mathematics, the small Veblen ordinal is a certain large countable ordinal, named after Oswald Veblen. It is occasionally called the Ackermann ordinal…
- Large countable ordinal In the mathematical discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non-circularl…
- Kleene's O In set theory and computability theory, Kleene's O {\displaystyle {\mathcal {O}}} is a canonical subset of the natural numbers when regarded as ordinal…
- Ordinal collapsing function In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining (notations for) certain recursive…
- Veblen function In mathematics, the Veblen functions are a hierarchy of normal functions (continuous strictly increasing functions from ordinals to ordinals), introduced…
- Fixed-point lemma for normal functions The fixed-point lemma for normal functions is a basic result in axiomatic set theory stating that any normal function has arbitrarily large fixed points…
- Additively indecomposable ordinal In set theory, a branch of mathematics, an additively indecomposable ordinal α is any ordinal number that is not 0 such that for any β , γ < α {\displaystyle…