Timeline for Where in ordinary math do we need unbounded separation and replacement?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Jul 22, 2021 at 2:15 | comment | added | David Roberts♦ | Friedman showed, and others refined, that you need iterations of the powerset construction countable-ordinal-many times, applied to the game tree in question. Replacement is usually used to construct these, but it's a very localised usage. Starting from a Polish space, and payoff from a Borel set therein, I believe it's enough to ask that $\beth_\alpha$ exists for all countable ordinals $\alpha$. | |
| Feb 11, 2013 at 19:15 | vote | accept | Andrej Bauer | ||
| Feb 11, 2013 at 19:15 | |||||
| Feb 11, 2013 at 6:01 | comment | added | Andrej Bauer | Well, let's not fret too much about what is ordinary. I am just trying to avoid having listed obvious stuff of the sort "there is life beyond $V_{\omega + \omega}$". I do think Borel determinacy is border-line (descriptive) set theory, but the cool thing is that replacement creeps up in a non-obvious way. | |
| Feb 11, 2013 at 2:28 | comment | added | François G. Dorais | Unimportant or underused... This one is a tough call! | |
| Feb 11, 2013 at 1:51 | comment | added | Todd Trimble | Hm, I'd call it a theorem of descriptive set theory with applications that are occasionally useful to workers in real analysis. (Maybe you could place it on the border of real analysis, but not necessarily in the interior (-: .) Such quibbles aside, it seems to have "ordinary mathematics" content. | |
| Feb 11, 2013 at 0:47 | comment | added | arsmath | Borel determinacy is a theorem in real analysis, and therefore part of "ordinary mathematics". Maybe it's not an important theorem in real analysis, but it is indisputably a theorem in real analysis. | |
| Feb 10, 2013 at 22:55 | comment | added | user30035 | But Borel Determinacy is surely a great example of the sort of thing that is known and loved by logicians but which is never used in "ordinary mathematics" (by which I mean e.g. "the things talked about over the last 20 years in the Harvard number theory seminar"). So if this is what you are looking for then perhaps I've misunderstood the question? | |
| Feb 10, 2013 at 21:08 | comment | added | Andrej Bauer | That's what I am looking for! Thanks. | |
| Feb 10, 2013 at 18:13 | history | answered | Cody Dance | CC BY-SA 3.0 |