Timeline for Is the Ordering Principle equivalent to a selection principle?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Dec 26, 2023 at 17:24 | comment | added | Joel David Hamkins | Ah, yes, that's what the argument gives, since we are splitting each equivalence class in two at each step. And the converse is immediate. Great! | |
| Dec 26, 2023 at 17:15 | comment | added | François G. Dorais | Yes, this is known as the Kinna-Wagner Selection Principle. It is actually equivalent to: every set injects into $2^\alpha$ for some ordinal $\alpha$. (IIRC this is in Jech's The Axiom of Choice.) Since $2^\alpha$ is linearly ordered, the Ordering Principle follows. | |
| Dec 25, 2023 at 23:20 | comment | added | Joel David Hamkins | I made a tweet explaining this latter point. (Perhaps this is already known?) twitter.com/JDHamkins/status/1739421469426266335 | |
| Dec 25, 2023 at 21:22 | comment | added | Joel David Hamkins | For example, it suffices to pick nontrivial subsets of a given set, since such a choice function gives a nontrivial preorder. | |
| Dec 25, 2023 at 21:04 | comment | added | Joel David Hamkins | Nice! You iteratively apply it to the equivalence classes to gradually resolve the whole order. | |
| S Dec 25, 2023 at 20:03 | history | answered | François G. Dorais | CC BY-SA 4.0 | |
| S Dec 25, 2023 at 20:03 | history | made wiki | Post Made Community Wiki by François G. Dorais |