A natural number (which, in this context, includes the number 0) can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. When restricted to finite sets, these two concepts

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Well-ordered sets
In a well-ordered set, every non-empty subset contains a distinct smallest element. Given the axiom of dependent choice, this is equivalent to saying that the set is totally orderedand there is no infinite decreasing sequence (the

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If α is any ordinal and X is a set, an α-indexed sequence of elements of X is a function from α to X. This concept, a transfinite sequence (if α is infinite) or ordinal-indexed sequence,

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Transfinite induction holds in any well-ordered set, but it is so important in relation to ordinals that it is worth restating here.
Any property that passes from the set of ordinals smaller than a given ordinal α to α itself, is true of all ordinals

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Initial ordinal of a cardinal
Each ordinal associates with one cardinal, its cardinality. If there is a bijection between two ordinals (e.g. ω = 1 + ω and ω + 1 > ω), then they

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As mentioned above (see Cantor normal form), the ordinal ε0 is the smallest satisfying the equation , so it is the limit of the sequence 0, 1, , , , etc. Many ordinals can be defined in such a

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There are three usual operations on ordinals: addition, multiplication, and (ordinal) exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set that represents the operation or by using transfinite

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