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Order isomorphism

  1. Dec 26, 2011 #1


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    I am trying to prove the following results: If α and β are ordinals, then the orderings (α,∈) and (β,∈) are isomorphic if and only if α = β.

    So far, I have only proved that the class Ord is transitive and well-ordered by ∈. I can prove this result with the following lemma: If f:α→β is an order-preserving map, then z ≤ f(z). However, I am having difficulty proving this lemma without something like transfinite induction. Any help?
  2. jcsd
  3. Dec 26, 2011 #2
    Assume that [itex]z\leq f(z)[/itex] does NOT hold. Then there is a least a such that [itex]f(a)<a[/itex]. Take f of both sides.
  4. Dec 26, 2011 #3


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    That works perfectly! I am silly for not thinking of something like that. Thanks!
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