Timeline for Which ordinals can be embedded into an ordered field?
Current License: CC BY-SA 3.0
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| Mar 4, 2016 at 18:53 | comment | added | Philip Ehrlich | For a striking illustration of Joel’s observation, let $\mathbf{No}$ be the ordered field of surreal numbers, $a$ be an indeterminate where $a>\mathbf{No}$ and $\mathbf{No}(a)$ be the ordered simple transcendental extension of $\mathbf{No}$ generated by $a$ and $\mathbf{No}$. Although $\mathbf{No}(a)$ has cofinality $\omega$, it contains the entire class of ordinals. This ordered field, which is constructible in NBG, is discussed in the author’s JSL (2001: Proposition 3, p. 1240) for reasons unrelated to the question at hand. | |
| Mar 4, 2016 at 17:19 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| Mar 4, 2016 at 17:11 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| Mar 4, 2016 at 16:59 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |