Timeline for Embedding large countable ordinals into the complex plane
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Sep 17, 2023 at 16:04 | vote | accept | 0x11111 | ||
| Sep 14, 2023 at 14:58 | comment | added | Joel David Hamkins | The computable ordinals are also exactly those that support a computable copy as a subset of the rationals. | |
| Sep 14, 2023 at 14:46 | comment | added | Joel David Hamkins | If one is thinking of computability, then you will land at the Church-Kleene ordinal, the supremum of all ordinals representable by a computable relation on the natural numbers. en.wikipedia.org/wiki/Nonrecursive_ordinal | |
| Sep 14, 2023 at 14:42 | comment | added | მამუკა ჯიბლაძე | Fair enough. No, I cannot formulate any rigorous requirement. Actually I am rather wondering how would an obvious failure look like. Let me say it this way: surely one cannot do it for $\omega_1$ as no explicit well-ordering of the continuum is possible in any sense, right? So one might expect the same for some countable ordinals "so big they are as bad as $\omega_1$"... | |
| Sep 14, 2023 at 11:09 | comment | added | Joel David Hamkins | The embedding is an isomorphism to a closed countable well-ordered set of rational numbers. This makes them quite explicit by many accounts of what this word means. (For example, the world of Borel sets and functions is sometimes described as the realm of explicit mathematics, and closed sets are Borel.) If you want something more explicit than this, then I'd want you to state clearly what the requirements are. | |
| Sep 14, 2023 at 5:09 | comment | added | მამუკა ჯიბლაძე | But if one asks about an explicit embedding? Presumably embeddings if non-recursive ordinals cannot be explicitly described, or? | |
| Sep 14, 2023 at 4:13 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
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| Sep 14, 2023 at 4:01 | history | answered | Joel David Hamkins | CC BY-SA 4.0 |