Webb14 okt. 2024 · Lemma 1: Suppose that ( W, <) is well-ordered and that I is an initial segment of W. Then I = { x ∈ W ∣ x < a } for a unique a ∈ W. Proof: It is easy to verify … Webb24 mars 2024 · Initial Segment Let be a well ordered set. Then the set for some is called an initial segment of (Rubin 1967, p. 161; Dauben 1990, pp. 196-197; Moore 1982, pp. …
Chapter VIII Ordered Sets, Ordinals and Transfinite Methods
Webborder-types of well-ordered sets. Theorem 2.12. Every well-ordered set is isomorphic to a unique ordinal number. Proof. The uniqueness follows from Lemma 2.7. Given a well-ordered set W, we find an isomorphic ordinal as follows: DefineF(x)=α if α is isomorphic to the initial segment of W given by x.Ifsuchanα exists, then it is unique. Webb26 nov. 2024 · A well ordered set is a totally ordered set in which every non-empty subset has a smallest element. The interval [ 0, 1] ⊆ R with the usual ordering has a smallest element but is not well ordered because (among other reasons) the subset ( 0, 1) has no smallest element. – Robert Shore Nov 26, 2024 at 1:25 What about the well … ibew236
Well-ordered set - Encyclopedia of Mathematics
Webb24 Chapter 6: Ordered sets Problems The ordered set A,≺ is called well ordered if every nonempty subset contains a smallest element, i.e., if for every X ⊆A, X =∅thereisana ∈X such that for every b ∈X we have a b. If A,≺ is an ordered set, then X ⊆A is an initial segment if a ∈X and b ≺a imply b ∈X (intuitively, X consists of a starting section of … Webb9 juli 2011 · However, Lindenbaum proved the following curious fact (which is true in any cardinality): if a linear ordering A is isomorphic to an initial segment (downwards closed set, ideal) of a linear ordering B, and B is isomorphic to a final segment (upwards closed) of A, then A and B are isomorphic. Webbordered sets declaring two well ordered sets to be equivalent if they are isomorphic in the same way we de ned cardinal numbers. The resulting equivalence classes are … monash graduate research