Timeline for Exponentiation and Dedekind-finite cardinals
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Aug 26, 2014 at 22:00 | comment | added | Asaf Karagila♦ | [...] For example, list all the names in the ground model, and then map each name into a pair of countable ordinals $(\alpha,\beta)$, such that there is an antichain interpreting the name as an actual symmetric name, and then mapping it into \beta. You can do that, then choose equivalence classes under orbits of permutations (which move finitely many reals anyway) and create a name which is stable under automorphisms. By counting arguments you don't go above $\omega_1$ names, so everything is just peachy. | |
| Aug 26, 2014 at 21:56 | comment | added | Asaf Karagila♦ | Andres, here is a sketch, and you can tell me what's wrong with it. Let $\dot A$ be the name of the generic D-finite set. Then there is a name (for the full extension) of an injection from $2^A$ into $2^\omega$, since in the full Cohen model, $A$ is countable. Now consider a list in the ground model of all the names of subsets of $A$. Since the forcing is ccc, there are only continuum many, so we there are only continuum many symmetric ones (externally). Now by usual arguments build a name which $1$ forces to be an injection from all these sets into $\omega_1$. [...] | |
| Aug 26, 2014 at 17:09 | comment | added | Asaf Karagila♦ | Yeah, that is also true. I'll give it some thought, I'm sure there's a quick argument. | |
| Aug 26, 2014 at 17:00 | comment | added | Andrés E. Caicedo | The other direction is the key, any D - infinite power set is at least as large as the reals. | |
| Aug 26, 2014 at 16:49 | comment | added | Asaf Karagila♦ | Andres, in the Cohen model, I believe the power set of the Dedekind finite set of reals should be equal to the reals. It is certainly as big. I'll have to think about an argument for the other direction, but I'm sure it's true. | |
| Aug 26, 2014 at 16:41 | comment | added | Andrés E. Caicedo | Asaf, do you know if the power set of a D - finite set can be equipotent to the set of reals? If not, how about its double power set, and the power set of the reals? If yes, what if in addition we ask that the D - finite set be itself a power set? (Is there a reference for these themes?) (I owe you an email. Please give me a few hours.) | |
| Aug 26, 2014 at 13:41 | history | answered | Asaf Karagila♦ | CC BY-SA 3.0 |