Timeline for What are some examples of proving that a thing exists by proving that the set of such things has positive measure?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Jul 15, 2020 at 8:02 | comment | added | Nicola Gigli | @user21820: "It is unlikely that Terry's note is an unpacking ..." yeah, that was my bet as well | |
| Jul 15, 2020 at 7:51 | comment | added | user21820 | @NicolaGigli: Sorry there was a typo; the "$∃m,c{∈}N$" should be "$∃m{\in}N_{>k}\ ∃c{∈}N$". It is unlikely that Terry's note is an unpacking of the full Shoenfield absoluteness, since Shoenfield did it for sentences higher in the analytical hierarchy, whereas here we only need it for a $Π^1_1$ sentence. I believe there is a much simpler proof for this special case, but haven't really thought about it. | |
| Jul 15, 2020 at 7:49 | comment | added | Nicola Gigli | @user21820: thank you, I'll think about this. Still, I think it is interesting to have a more direct path like the one in Terence Tao's note (whether this can be regarded as an unpacking of Shoenfield's theorem in this particular circumstance, I have no idea) | |
| Jul 15, 2020 at 7:41 | comment | added | user21820 | @NicolaGigli: I am not a set theory expert, but from what I know AC can be eliminated from any ZFC proof of any theorem that is $Σ^1_2$ or $Π^1_2$, by simply applying Shoenfield's absoluteness theorem. In particular, the statement $Q$ here is equivalent to a $Π^1_1$ sentence that says "$∀A{⊆}N\ $ $( \ ∃p,q{∈}N\ ∀k{∈}N\ ∃m,c{∈}N\ ∃t{∈}(A⋂[-m,m])^c ( \ p,q>0 ∧ c·q > (2·m+1)·p \ )$ $⇒ ∀c{∈}N\ ∃t{∈}A^c\ ( \ \text{$t$ is an AP} \ ) \ )$" where $N$ is the naturals. | |
| Jul 15, 2020 at 7:06 | comment | added | Nicola Gigli | @user21820: I don't really know, it is not my field of expertise. I've just seen the use of Hahn-Banach on a non separable space in the argument above and wondered whether more than DC was truly needed to carry it out. It might be that one could answer `no' by some general principle of which I known nothing about (but I'd like to know more: can you explain or point me to some relevant literature?) | |
| Jul 15, 2020 at 6:20 | comment | added | user21820 | @NicolaGigli: Doesn't non-dependence on AC follow from $Π^1_1$-absoluteness? | |
| Jul 15, 2020 at 5:16 | comment | added | Nicola Gigli | Very interesting, thank you very much | |
| Jul 15, 2020 at 0:04 | comment | added | Terry Tao | Actually I now remember that I looked into this question back in 2005 and found a version of the Furstenberg correspondence principle that does not require the axiom of choice: math.ucla.edu/~tao/preprints/Expository/limiting.pdf | |
| Jul 14, 2020 at 23:46 | comment | added | Terry Tao | There is another way to proceed with the derivation by constructively building some finite probability measures on a Cantor space that approximately model the set $A$ and then extracting a weakly convergent subsequence. I think this has a chance of being done completely in a choice-free manner (countable iterations of Bolzano-Weierstrass plus a diagonalization argument). Not an expert on these questions though. [And now one can go further and ask if one can make this argument intuitionistic, my guess is no...] | |
| Jul 14, 2020 at 23:17 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Jul 14, 2020 at 21:36 | comment | added | Nicola Gigli | a curiosity: is it really necessary for this argument to work to have at disposal the (weak version of - but still stronger than Countable Dependent) Axiom of Choice? I ask because I don't think anything more than DC is needed to prove both Furstenberg's theorem and Szemerédi's one (but I might be wrong here), thus I find curious that some `serious choice' is needed to bridge them | |
| Jul 14, 2020 at 20:31 | history | edited | Terry Tao | CC BY-SA 4.0 |
added 4 characters in body
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| Jul 14, 2020 at 20:25 | history | edited | Tom Leinster | CC BY-SA 4.0 |
fixed typo
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| Jul 14, 2020 at 20:22 | history | answered | Terry Tao | CC BY-SA 4.0 |