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Results tagged with ordinal-numbers
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user 111429
An ordinal is the order type of a well-ordered set. The first few ordinals are $0, 1, 2, \dots, \omega, \omega+1, \dots$ where $\omega$ is the order type of $\mathbb{N}$, and $\omega+1$ is the order type of $\mathbb{N}$ together with a maximum element.
6
votes
3
answers
1k
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Do these ordinals exist?
Given an ordinal $\alpha$, I define $F_{n}(\alpha)$ as follows:
$F_0(\alpha)=\alpha$
$F_{n+1}(\alpha)$ is the smallest $\beta$ such that no first-order $\phi$ in the
language of $\{\in\}$ has $(\ma …
- 675
1
vote
Do these ordinals exist?
You can guarantee $F_n(\alpha)$ is countable.
Assume the contrary. There is a first-order formula for every countable ordinal $\phi$ such that $(V\models\phi(S,F_1(\alpha)...))\Leftrightarrow S=\alp …
- 675