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Mathematical technique used in proof theory
In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength.
If theories have the same proof-theoretic ordinal they are often equiconsistent, and if one theory has a larger proof-theoretic ordinal than another it can often prove the consistency of the second theory.
In addition to obtaining the proof-theoretic ordinal of a theory, in practice ordinal analysis usually also yields various other pieces of information about the theory being analyzed, for example characterizations of the classes of provably recursive, hyperarithmetical, or functions of the theory.[1]
^M. Rathjen, "Admissible Proof Theory and Beyond". In Studies in Logic and the Foundations of Mathematics vol. 134 (1995), pp.123--147.
In proof theory, ordinalanalysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories...
In statistics, ordinal regression, also called ordinal classification, is a type of regression analysis used for predicting an ordinal variable, i.e....
of the major areas of proof theory include structural proof theory, ordinalanalysis, provability logic, reverse mathematics, proof mining, automated theorem...
have computable ordinal notations (see ordinalanalysis). However, it is not possible to decide effectively whether a given putative ordinal notation is a...
Other examples of ordinal data include socioeconomic status, military ranks, and letter grades for coursework. Ordinal data analysis requires a different...
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non-recursive ordinals are large countable ordinals greater than all the recursive ordinals, and therefore can not be expressed using recursive ordinal notations...
definition of ordinal collapsing functions is inextricably intertwined with the theory of ordinalanalysis, since the large countable ordinals defined and...
have an ordinal notation in Kleene's O {\displaystyle {\mathcal {O}}} . Arithmetical hierarchy Large countable ordinalOrdinalanalysisOrdinal notation...
Lane. The Howard ordinal (also known as the Bachmann–Howard ordinal) was named after him. He was the first to carry out an ordinalanalysis of the intuitionistic...
enumerating function can assume any ordinal. Under this definition, an enumeration of a set S is any surjection from an ordinal α onto S. The more restrictive...
this ordering (it is in fact the least ordinal with this property, and as such, in proof-theoretic ordinalanalysis, is used as a measure of the strength...
best-known classification with four levels, or scales, of measurement: nominal, ordinal, interval, and ratio. This framework of distinguishing levels of measurement...
in this manner, it is possible to define a cardinal number ℵα for every ordinal number α, as described below. The concept and notation are due to Georg...
inclusion. The ordinal numbers are a simple example: if each ordinal n is identified with the set [ n ] {\displaystyle [n]} of all ordinals less than or...
list Complete theory Independence (from ZFC) Proof of impossibility Ordinalanalysis Reverse mathematics Self-verifying theories Model theory Interpretation...
recursion) an ordinal number α {\displaystyle \alpha } , known as its rank. The rank of a pure set X {\displaystyle X} is defined to be the least ordinal that...
in mathematical analysis: a domain is a non-empty connected open set in a topological space. In particular, in real and complex analysis, a domain is a...
which are followed by the aleph numbers. The aleph numbers are indexed by ordinal numbers. If the axiom of choice is true, this transfinite sequence includes...
In the von Neumann construction of the ordinals, 0 is defined as the empty set, and the successor of an ordinal is defined as S ( α ) = α ∪ { α } {\displaystyle...
designated for each equivalence class. The most common choice is the initial ordinal in that class. This is usually taken as the definition of cardinal number...
function f ( x ) = x 2 {\displaystyle f(x)=x^{2}} as it is used in real analysis (that is, as a function that inputs a real number and outputs its square)...
is the third largest city in the country"), in which case they serve as ordinal numbers. Natural numbers are sometimes used as labels—also known as nominal...
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