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Classification of Numbers [number theory]
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cardinals [wiki]
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ordinals (proper class) [wiki]
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Generalization of natural numbers for describing sizes of infinite sets.
The generalized continuum hypothesis (GCH) states that
for every infinite set X,
there are no cardinals strictly between |X| and 2|X|.
«A cardinal is called a large cardinal
if it belongs to a class of cardinals the existence of which
(provably) cannot be proved within ZFC.»
See [wiki] for examples.
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Generalization of natural numbers for describing positions within infinite sequences(?)
John von Neumann's definition:
«A set S is an ordinal if and only if S is totally ordered with respect to set containment and every element of S
is also a subset of S.»
Every well-ordered set is order-isomorphic to exactly one of these ordinals.
Every ordinal S is a set having as elements precisely the ordinals smaller than S.
Every set of ordinals is well-ordered.
Every set of ordinals has a supremum, the ordinal gotten by taking the union of all the ordinals in the set.
The collection of all ordinals is not a set (ie., is a proper class).
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«A natural number can be used for two purposes: to describe the size of a set, or to describe the position of an element in a
sequence. While in the finite world these two concepts coincide, when dealing with infinite sets one has to distinguish
between the two. The size aspect leads to cardinal numbers, which were also discovered by Cantor, while the position
aspect is generalized by the ordinal numbers...»
[wiki]
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| N - naturals numbers
| | = closure of 1 (lately more commonly of 0) under successor and/or addition
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| Z - integral numbers
| | = closure of N under subtraction
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| Q - rational numbers
| | = closure of Z under non-zero division
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| A - algebraic numbers
| = closure of Q under integral roots [wiki]
"algebraic closure" of 1, ie., closure under +, -, *, / and integral roots
A is a field (ie. closed under +-*/),
more precisely, algebra (A, +, *) is a field[^]
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not constructively defined
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| computable numbers
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- is an algebraically closed field (+-*/ and integral root closed),
but not a recursive set [wiki]
E.g.,  , e
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definable numbers
| - is a countable, incomplete field [wiki]
A defineable but uncomputable number: Chaitin's constant 
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R - real numbers
| | - is an overcountable, complete ordered field
Sorry, no example of an undefinable number can be define here.
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*R - hyperreal numbers (aka. non-standard reals) [wiki]
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= reals
+ infinitely large numbers
+ infinitesimal numbers
Is an ordered field; not metric space but order topology; suspected to be nonconstructive.
«The hyperreals are defined in such a way that every first-order logic statement that uses basic arithmetic (the natural
numbers, plus, times, comparison) and quantifies only over the real numbers is also true if we presume that they quantify
over hyperreal numbers.»
«When Isaac Newton and Gottfried Leibniz introduced differentials,
they used infinitesimals and these were still regarded as useful by
Leonhard Euler and Augustin Louis Cauchy.»
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surreal numbers [wiki]
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= reals
+ "infinite" numbers which are larger than any real number
+ "infinitesimal" numbers that are closer to zero than any real number
+ each real number is surrounded by surreals that are closer to it than any real number.
Is a field and proper class [wiki]
«[F]irst proposed by John Conway and later detailed by Donald Knuth» in 1974.
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