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Results tagged with order-theory
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user 1004
18
votes
Accepted
Order type of the smallest set containing the identity function and closed under exponentiation
As Joel showed, the set $E$ is well-ordered with order type no more than the Cantor ordinal $\epsilon_0$. In fact, its order type is exactly $\epsilon_0$. This can be proved by constructing the order …
- 17.1k
15
votes
Accepted
Converse to Banach’s fixed point theorem for ordered fields?
Yes, it is true that $R$ must be the field of real numbers.
As $R$ is an ordered field, it is naturally an extension $\mathbb{Q}\hookrightarrow R$. We can prove the following two properties, which ch …
- 17.1k