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When considered as a set, the elements of are the countable ordinals (including finite ordinals), [1] of which there are uncountably many. Like any ordinal number (in von Neumann's approach ), ω 1 {\displaystyle \omega _{1}} is a well-ordered set , with set membership serving as the order relation.
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 ...
The set S together with the ordering is then called a well-ordered set or woset. [1] In some academic articles and textbooks these terms are instead written as wellorder, wellordered, and wellordering or well order, well ordered, and well ordering. Every non-empty well-ordered set has a least element.
If admits a totally ordered cofinal subset, then we can find a subset that is well-ordered and cofinal in . Any subset of is also well-ordered. Two cofinal subsets of with minimal cardinality (that is, their cardinality is the cofinality of ) need not be order isomorphic (for example if = +, then both + and {+: <} viewed as subsets of have the countable cardinality of the cofinality of but are ...
A concept in set theory and logic that categorizes well-ordered sets by their structure, such that two sets have the same order type if there is a bijective function between them that preserves order. ordinal 1. An ordinal is the order type of a well-ordered set, usually represented by a von Neumann ordinal, a transitive set well ordered by ∈. 2.
A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering. The well-ordering theorem together with Zorn's lemma are the most important mathematical statements that are equivalent to the axiom of choice (often called AC, see also Axiom of choice § Equivalents).
We'll cover exactly how to play Strands, hints for today's spangram and all of the answers for Strands #287 on Sunday, December 15. Related: 16 Games Like Wordle To Give You Your Word Game Fix ...
In mathematics, specifically in axiomatic set theory, a Hartogs number is an ordinal number associated with a set. In particular, if X is any set, then the Hartogs number of X is the least ordinal α such that there is no injection from α into X. If X can be well-ordered then the cardinal number of α is a minimal cardinal greater than that of X.