In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories...

of the major areas of proof theory include structural proof theory, ordinal analysis, provability logic, reverse mathematics, proof mining, automated theorem...

have computable ordinal notations (see ordinal analysis). 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...

In statistics, ordinal regression, also called ordinal classification, is a type of regression analysis used for predicting an ordinal variable, i.e....

definition of ordinal collapsing functions is inextricably intertwined with the theory of ordinal analysis, since the large countable ordinals defined and...

have an ordinal notation in Kleene's O {\displaystyle {\mathcal {O}}} . Arithmetical hierarchy Large countable ordinal Ordinal analysis Ordinal notation...

Fan; Jöreskog, Karl G.; Luo, Hao (2010-07-13). "Confirmatory Factor Analysis of Ordinal Variables With Misspecified Models". Structural Equation Modeling...

non-recursive ordinals are large countable ordinals greater than all the recursive ordinals, and therefore can not be expressed using recursive ordinal notations...

best-known classification with four levels, or scales, of measurement: nominal, ordinal, interval, and ratio. This framework of distinguishing levels of measurement...

Lane. The Howard ordinal (also known as the Bachmann–Howard ordinal) was named after him. He was the first to carry out an ordinal analysis of the intuitionistic...

Cantor developed a theory of transfinite numbers, called cardinals and ordinals, which extended the arithmetic of the natural numbers. His notation for...

asserting that the ordinal called ε0 is wellfounded; see Gentzen's consistency proof. Gentzen's theorem spurred the development of ordinal analysis in proof theory...

trichotomous. A weakened theory of ordinals in turn affects the proof theoretic strength defined in ordinal analysis. In exchange, constructive set theories...

ISBN 978-1-77048-113-8. Carlson, Robert (2017). A Concrete Introduction to Real Analysis. CRC Press. ISBN 978-1-4987-7815-2. Cohen, Elliot D. (2009). Critical Thinking...

variables with more than two values there is the multinomial logit. For ordinal variables with more than two values, there are the ordered logit and ordered...

like "this is the third largest city in the country", which are called ordinal numbers. Natural numbers are also used as labels, like jersey numbers on...

proof-theoretic ordinal analysis. In ordinal analysis one gauges the strength of theories by measuring how large the (constructive) ordinals are that can...

initiated the program of ordinal analysis in proof theory. In this program, formal theories of arithmetic or set theory are assigned ordinal numbers that measure...

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)...

theory and proof theory, the Takeuti–Feferman–Buchholz ordinal (TFBO) is a large countable ordinal, which acts as the limit of the range of Buchholz's psi...

In mathematical logic and set theory, an ordinal notation is a partial function mapping the set of all finite sequences of symbols, themselves members...

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...

example, that no set is an element of itself and that every set has an ordinal rank. Subsets are commonly constructed using set builder notation. For...

list Complete theory Independence (from ZFC) Proof of impossibility Ordinal analysis Reverse mathematics Self-verifying theories Model theory Interpretation...

infinite cardinal number ℵ α {\displaystyle \aleph _{\alpha }} for every ordinal number α , {\displaystyle \alpha ,} as described below. The concept and...

1-CA0. Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes...

in professional ice hockey Proof-theoretic ordinal, a concept in ordinal analysis that refers to an ordinal assigned to a mathematical theories to measure...

state a paradox: the Burali-Forti paradox shows that the collection of all ordinal numbers cannot form a set. Very soon thereafter, Bertrand Russell discovered...

In mathematics, even and odd ordinals extend the concept of parity from the natural numbers to the ordinal numbers. They are useful in some transfinite...