Timeline for Adding a real with infinite conditions
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Jan 26, 2015 at 19:10 | comment | added | François G. Dorais | @Asaf The generalized Cohen forcing and generalized Silver forcing were introduced by Serge Grigorieff Combinatorics on ideals and forcing, Ann. Math. Logic 3 (1971), 363-394; MR0297560. | |
| Jan 26, 2015 at 17:04 | comment | added | 喻 良 | Jockusch told me that a similar method (restrict the conditions with the recursive ones) was also used by Spector, and later by Lachlan, to construct minimal Turing degrees. | |
| Jan 26, 2015 at 14:17 | comment | added | Asaf Karagila♦ | Yes, I will definitely give this more thought. But first I really really really gotta finish banging some argument shut about choiceless forcing (and actually the question here is somewhat related to trying some alternative approach in solving the issue I'm dealing with by changing the forcing a bit). I think I'll drag a couple other Ph.D. students with me to these sort of questions. | |
| Jan 26, 2015 at 14:14 | comment | added | Joel David Hamkins | I'd encourage you to go for it! And what of other natural ideals, such as the ideal of all $A\subset\omega$ for which $\sum_{n\in A}\frac1n$ converges, or the ideal of all asymptotic density zero sets? | |
| Jan 26, 2015 at 14:10 | comment | added | Asaf Karagila♦ | I'm sure that it's possible to investigate properties of the new sets inherited from the ideal and the choice of zero or co-positive domains, in a context much more general than ideals on $\omega$ (but I do expect things to get much more complicated, of course... and depend on additional hypotheses like cardinal arithmetics and large cardinals). | |
| Jan 26, 2015 at 14:09 | comment | added | Asaf Karagila♦ | Looking at Jech's book it strikes me that we really have two tricks up our sleeve for adding a new subset to some $X$. Fix some ideal on $X$ and forcing with partial functions from $X$ to $2$; either the domains are from the ideal, or the domains are co-positive sets. The ideal of finite sets gives us Cohen in the first case and Prikry-Silver in the latter case. | |
| Jan 26, 2015 at 13:18 | vote | accept | Asaf Karagila♦ | ||
| Jan 26, 2015 at 13:15 | comment | added | Asaf Karagila♦ | Aha! I knew I'd seen it before. And just as luck would have it, I have a copy of that book from the library with me. I wonder why I didn't check it before! | |
| Jan 26, 2015 at 13:14 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |