von Neumann ordinal - definizione. Che cos'è von Neumann ordinal
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Cosa (chi) è von Neumann ordinal - definizione

MATHEMATICAL CONCEPT
Initial ordinal; Von Neumann cardinal

von Neumann ordinal         
  • A graphical "matchstick" representation of the ordinal ω². Each stick corresponds to an ordinal of the form ω·''m''+''n'' where ''m'' and ''n'' are natural numbers.
ORDER TYPE OF A WELL-ORDERED SET
Ordinal numbers; Von Neumann ordinal; Ordinal Number; Ordinal (mathematics); Transfinite ordinal number; Transfinite ordinal numbers; Finite ordinal number; Ordinal number (finite); Transfinite sequence; Ω (ordinal number); Ordinal number (mathematics); O (ordinal number); Ordinal number (set theory); Least infinite ordinal; Second number class; First number class; Omega (set theory); Ω+1; First infinite ordinal; First infinite ordinal number; Countable ordinals; Countable ordinal; Von Neumann definition of ordinals; Von Neumann encoding; Number class; Omega (ordinal); Von Neumann ordinals
<mathematics> An implementation of ordinals in set theory (e.g. Zermelo Frankel set theory or ZFC). The von Neumann ordinal alpha is the well-ordered set containing just the ordinals "shorter" than alpha. "Reasonable" set theories (like ZF) include Mostowski's Collapsing Theorem: any well-ordered set is isomorphic to a von Neumann ordinal. In really screwy theories (e.g. NFU -- New Foundations with Urelemente) this theorem is false. The finite von Neumann ordinals are the {von Neumann integers}. (1995-03-30)
ordinal number         
  • A graphical "matchstick" representation of the ordinal ω². Each stick corresponds to an ordinal of the form ω·''m''+''n'' where ''m'' and ''n'' are natural numbers.
ORDER TYPE OF A WELL-ORDERED SET
Ordinal numbers; Von Neumann ordinal; Ordinal Number; Ordinal (mathematics); Transfinite ordinal number; Transfinite ordinal numbers; Finite ordinal number; Ordinal number (finite); Transfinite sequence; Ω (ordinal number); Ordinal number (mathematics); O (ordinal number); Ordinal number (set theory); Least infinite ordinal; Second number class; First number class; Omega (set theory); Ω+1; First infinite ordinal; First infinite ordinal number; Countable ordinals; Countable ordinal; Von Neumann definition of ordinals; Von Neumann encoding; Number class; Omega (ordinal); Von Neumann ordinals
¦ noun a number defining a thing's position in a series, such as 'first' or 'second'.
ordinal number         
  • A graphical "matchstick" representation of the ordinal ω². Each stick corresponds to an ordinal of the form ω·''m''+''n'' where ''m'' and ''n'' are natural numbers.
ORDER TYPE OF A WELL-ORDERED SET
Ordinal numbers; Von Neumann ordinal; Ordinal Number; Ordinal (mathematics); Transfinite ordinal number; Transfinite ordinal numbers; Finite ordinal number; Ordinal number (finite); Transfinite sequence; Ω (ordinal number); Ordinal number (mathematics); O (ordinal number); Ordinal number (set theory); Least infinite ordinal; Second number class; First number class; Omega (set theory); Ω+1; First infinite ordinal; First infinite ordinal number; Countable ordinals; Countable ordinal; Von Neumann definition of ordinals; Von Neumann encoding; Number class; Omega (ordinal); Von Neumann ordinals
(ordinal numbers)
An ordinal number or an ordinal is a word such as 'first', 'third', and 'tenth' that tells you where a particular thing occurs in a sequence of things. Compare cardinal number
.
N-COUNT

Wikipedia

Von Neumann cardinal assignment

The von Neumann cardinal assignment is a cardinal assignment that uses ordinal numbers. For a well-orderable set U, we define its cardinal number to be the smallest ordinal number equinumerous to U, using the von Neumann definition of an ordinal number. More precisely:

| U | = c a r d ( U ) = inf { α O N   |   α = c U } , {\displaystyle |U|=\mathrm {card} (U)=\inf\{\alpha \in \mathrm {ON} \ |\ \alpha =_{c}U\},}

where ON is the class of ordinals. This ordinal is also called the initial ordinal of the cardinal.

That such an ordinal exists and is unique is guaranteed by the fact that U is well-orderable and that the class of ordinals is well-ordered, using the axiom of replacement. With the full axiom of choice, every set is well-orderable, so every set has a cardinal; we order the cardinals using the inherited ordering from the ordinal numbers. This is readily found to coincide with the ordering via ≤c. This is a well-ordering of cardinal numbers.