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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.
Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set. The well-ordering theorem, which is equivalent to the axiom of choice, states that every set can be well ordered. If a set is well ordered (or even if it merely admits a well-founded relation), the proof technique of ...
One can (even without invoking the regularity axiom) show that the set of all natural numbers such that "{, …,} is well-ordered" is inductive, and must therefore contain all natural numbers; from this property one can conclude that the set of all natural numbers is also well-ordered.
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 ...
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 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).
A well-ordered set is a totally ordered set (an ordered set such that, given two distinct elements, one is less than the other) in which every non-empty subset has a least element. Equivalently, assuming the axiom of dependent choice , it is a totally ordered set without any infinite decreasing sequence — though there may be infinite ...
In set theory, a tree is a partially ordered set (T, <) such that for each t ∈ T, the set {s ∈ T : s < t} is well-ordered by the relation <. Frequently trees are assumed to have only one root (i.e. minimal element), as the typical questions investigated in this field are easily reduced to questions about single-rooted trees.