In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation...

In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite...

a finite set is the common ordinal number of all possible well-orderings of that set, and cardinal and ordinal arithmetic (addition, multiplication, power...

numbers were introduced by Georg Cantor in the context of ordinal arithmetic; they are the ordinal numbers ε that satisfy the equation ε = ω ε , {\displaystyle...

interpret a sufficient portion of arithmetic to make statements about ordinal notations. The proof-theoretic ordinal of such a theory T {\displaystyle...

an ordinal number α is the smallest ordinal number greater than α. An ordinal number that is a successor is called a successor ordinal. The ordinals 1...

counterexamples in topology. Epsilon numbers (mathematics) Large countable ordinal Ordinal arithmetic "Set Theory > Basic Set Theory (Stanford Encyclopedia of Philosophy)"...

the focus on countable ordinals, ordinal arithmetic is used throughout, except where otherwise noted. The ordinals described here are not as large as...

limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less...

particular, it is the proof theoretic ordinal of the subsystem Π 1 1 {\displaystyle \Pi _{1}^{1}} -CA0 of second-order arithmetic; this is one of the "big five"...

properties of ordinal numbers: each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent...

{\displaystyle \omega _{n}} ). Infinite initial ordinals are limit ordinals. Using ordinal arithmetic, α < ω β {\displaystyle \alpha <\omega _{\beta }}...

elementary function arithmetic (EFA), also called elementary arithmetic and exponential function arithmetic, is the system of arithmetic with the usual elementary...

the proof-theoretic ordinal of Peano arithmetic. PRA's proof theoretic ordinal is ωω, where ω is the smallest transfinite ordinal. PRA is sometimes called...

Feferman–Schütte ordinal (Γ0) is a large countable ordinal. It is the proof-theoretic ordinal of several mathematical theories, such as arithmetical transfinite...

Arithmetic is an elementary branch of mathematics that studies numerical operations like addition, subtraction, multiplication, and division. In a wider...

well-founded ordinals. Ordinal analysis was originated by Gentzen, who proved the consistency of Peano Arithmetic using transfinite induction up to ordinal ε0....

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

not qualifying as an ordinal notation. Large countable ordinals Ordinal arithmetic Ordinal analysis D. Madore, A Zoo of Ordinals (p.2). Accessed 25 October...

Transfinite number Well-founded induction Zorn's lemma J. Schlöder, Ordinal Arithmetic. Accessed 2022-03-24. It is not necessary here to assume separately...

arithmetic: addition, subtraction, multiplication, division and inequality. This allows an axiomatic construction of numbers and ordinal arithmetic,...

types, well-orders, ordinal numbers, ordinal arithmetic, and the Burali-Forti paradox according to which the collection of all ordinal numbers cannot be...

of an infinity of infinities. He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact...

smallest ordinal number greater than the ranks of all members of the set. In particular, the rank of the empty set is zero, and every ordinal has a rank...

of the surreals. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. It has also been...

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

called "primitive recursive arithmetic with the additional principle of quantifier-free transfinite induction up to the ordinal ε0", is neither weaker nor...

given by f (α) = 1 + α (see ordinal arithmetic). But f (α) = α + 1 is not normal because it is not continuous at any limit ordinal; that is, the inverse image...

the Guttman scale shown below in Table 2: Table 2. Data of the four ordinal arithmetic skill variables are hypothesized to form a Guttman scale The set profiles...

twentieth century. The Principia covered only set theory, cardinal numbers, ordinal numbers, and real numbers. Deeper theorems from real analysis were not...