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  2. Ordinal notation - Wikipedia

    en.wikipedia.org/wiki/Ordinal_notation

    A Gödel numbering is a function mapping the set of well-formed formulae (a finite sequence of symbols on which the ordinal notation function is defined) of some formal language to the natural numbers. This associates each well-formed formula with a unique natural number, called its Gödel number.

  3. Ordinal arithmetic - Wikipedia

    en.wikipedia.org/wiki/Ordinal_arithmetic

    In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set that represents the result of the operation or by using transfinite recursion .

  4. Ordinal number - Wikipedia

    en.wikipedia.org/wiki/Ordinal_number

    There are other modern formulations of the definition of ordinal. For example, assuming the axiom of regularity, the following are equivalent for a set x: x is a (von Neumann) ordinal, x is a transitive set, and set membership is trichotomous on x, x is a transitive set totally ordered by set inclusion, x is a transitive set of transitive sets.

  5. Ordinal indicator - Wikipedia

    en.wikipedia.org/wiki/Ordinal_indicator

    Also, the ordinal indicators should be distinguishable from superscript characters. The top of the ordinal indicators (i.e., the top of the elevated letter a and letter o) must be aligned [1] with the cap height of the font. The alignment of the top of superscripted letters a and o will depend on the font.

  6. Ordinal analysis - Wikipedia

    en.wikipedia.org/wiki/Ordinal_analysis

    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.

  7. Ordinal numeral - Wikipedia

    en.wikipedia.org/wiki/Ordinal_numeral

    Ordinal indicator – Character(s) following an ordinal number (used when writing ordinal numbers, such as a super-script) Ordinal number – Generalization of "n-th" to infinite cases (the related, but more formal and abstract, usage in mathematics) Ordinal data, in statistics; Ordinal date – Date written as number of days since first day of ...

  8. Von Neumann cardinal assignment - Wikipedia

    en.wikipedia.org/wiki/Von_Neumann_cardinal...

    Also, is the smallest uncountable ordinal (to see that it exists, consider the set of equivalence classes of well-orderings of the natural numbers; each such well-ordering defines a countable ordinal, and is the order type of that set), is the smallest ordinal whose cardinality is greater than , and so on, and is the limit of for natural ...

  9. Aleph number - Wikipedia

    en.wikipedia.org/wiki/Aleph_number

    For any ordinal α we have α ≤ ω α. In many cases ω α is strictly greater than α. For example, it is true for any successor ordinal: α + 1 < ω α+1 holds. There are, however, some limit ordinals which are fixed points of the omega function, because of the fixed-point lemma for normal functions. The first such is the limit of the sequence