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.(adsbygoogle = window.adsbygoogle || []).push({});

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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Order isomorphism

Loading...

**Physics Forums | Science Articles, Homework Help, Discussion**