Timeline for What do models where the CH is false look like?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 8, 2013 at 18:59 | comment | added | David Fernandez-Breton | (continued) What you can do, following Cohen's work, is start with (say) the 23rd Woodin cardinal $\kappa$ and add $\kappa$ many Cohen reals. After doing that, we live in a new model of set theory where $2^{\aleph_0}=\kappa$, but also $\kappa$ is no longer a Woodin cardinal and, in fact, not even strongly strongly inaccessible. | |
| Dec 8, 2013 at 18:58 | comment | added | David Fernandez-Breton | There's a little mistake in what you said: the continuum cannot be the 23rd Woodin cardinal. It in fact cannot be any strongly inaccessible cardinal, because the very definition of strongly inaccessible prevents it (if $\kappa$ is strongly inaccessible then $\mathfrak c=2^{\aleph_0}<\kappa$ because $\aleph_0<\kappa$). | |
| Oct 16, 2009 at 21:33 | history | edited | John Goodrick | CC BY-SA 2.5 |
Made the statement of Cohen's Theorem more precise (since maybe we shouldn't assume ZFC is conssitent!).
|
| Oct 16, 2009 at 20:42 | comment | added | John Goodrick | You're right, thanks; and I have no idea who first specified just the possible values of the continuum, but certainly it was a bit before Shelah's time. I've edited my answer accordingly. | |
| Oct 16, 2009 at 20:41 | history | edited | John Goodrick | CC BY-SA 2.5 |
Minor edit in the reference to Easton's Theorem
|
| Oct 16, 2009 at 20:18 | comment | added | Eric Wofsey | Actually, Easton's Theorem is much stronger than just specifying the value of the continuum; it specifies 2^\kappa for all regular \kappa simultaneously. I'm certain the result for just the continuum was known before that; if it wasn't in Cohen's original work it was discovered shortly afterward. | |
| Oct 16, 2009 at 20:01 | history | answered | John Goodrick | CC BY-SA 2.5 |