Timeline for A strange planar set and the Continuum Hypothesis
Current License: CC BY-SA 3.0
10 events
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| Mar 5, 2018 at 12:03 | comment | added | Rahman. M | There is a mean called Davies trees which could enable you to remove CH. One reason in which usually we cannot go up beyond \aleph_1 when CH fails is that we cannot get a countable elementary substructure at stage \omega_1 if of course a proof using countable elemenatry substructures exists. Now, everything here is definable and you may use elementary substructure in order to get M under CH. Now Davies tree gives you again a countable elementary substructure at level \omega_1. You can take a look at the following interesting paper. arxiv.org/abs/1705.06195 | |
| Feb 28, 2018 at 10:43 | comment | added | Stefan Mesken | @David I have a candidate for that: $\mathrm{CH}$ holds if and only if the plane $\mathbb R^2$ can be covered by 3 clouds. (See here.) With a little more work you can actually squeeze out a characterization of $2^{\aleph_0} = \aleph_n$ for all $n < \omega$ -- a result we rediscovered over lunch during least year's Arctic Set Theory Conference. | |
| Feb 27, 2018 at 21:06 | comment | added | David Feldman | May I confess my pedagogical motivation: to devise the most elementary sounding statement I could that depends on CH for its proof. | |
| Feb 26, 2018 at 5:54 | comment | added | David Feldman | Not something I feel comfortable doing in ZFC :) | |
| Feb 26, 2018 at 5:46 | comment | added | Asaf Karagila♦ | Assume MA, of course! That's what you do. :) | |
| Feb 26, 2018 at 3:39 | comment | added | David Feldman | Well say you're looking to make $p$ the midpoint of $\alpha$ and $\beta$. One strategy would involve focusing the search on a certain circle $O$ centered at $p$. But then each pair of points already in $M$ eliminates $c$ points of $O$ from consideration. If there are more than countably many such pairs, you can't make an appeal to probability theory, so what would you do? | |
| Feb 26, 2018 at 3:34 | comment | added | Joel David Hamkins | Can't you just perform your recursion in length $\frak{c}$ rather than length $\omega_1$? This will avoid the need for CH. | |
| Feb 26, 2018 at 3:22 | history | edited | David Feldman | CC BY-SA 3.0 |
added 1 character in body
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| Feb 26, 2018 at 3:20 | history | edited | Martin Sleziak |
added (continuum-hypothesis) tag
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| Feb 26, 2018 at 3:08 | history | asked | David Feldman | CC BY-SA 3.0 |