Timeline for Producing finite objects by forcing!
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Dec 31, 2016 at 9:13 | comment | added | bof | But the question was about using forcing to prove the existence of a finite object. I'm not sure that a Turing degree, or even a finite set of Turing degrees, counts as a finite object. | |
| Dec 16, 2014 at 23:53 | comment | added | Benjamin Dickman | w.r.t. @AndreasBlass' comment ("As for making 'genuine' priority arguments look like forcing, I think there have been many attempts over the years, but I haven't really learned about any of them") I asked a related question about the priority method and forcing some time ago: mathoverflow.net/q/124011/22971 | |
| Dec 16, 2014 at 15:05 | comment | added | Joel David Hamkins | Yes, you are right. I am hoping for too much. | |
| Dec 16, 2014 at 15:03 | comment | added | Andreas Blass | Wouldn't your argument imply that every ill-founded recursive tree has a hyperarithmetical path? And doesn't that contradict a theorem of Kleene? I think the best you can expect in general is a path recursive in a complete $\Pi^1_1$ set. (In the case at hand, though, the Kleene-Post construction seems to produce $\Delta^0_2$ examples.) | |
| Dec 16, 2014 at 14:50 | comment | added | Joel David Hamkins | @AndreasBlass It seems to me that we do get absoluteness into the hyperarithmetic realm, since the assertion is $\Sigma^1_1$, and so the tree of attempts to build the example is ill-founded in $V$ and hence must be ill-founded in $L_{\omega_1^{CK}}$. So there will be an instance there. Doesn't this line of argument succeed? | |
| Dec 15, 2014 at 17:20 | comment | added | Joel David Hamkins | Yes, something like that is what I was asking about. I don't know any way to do anything like that. Perhaps one might more reasonably hope for a version of Shoenfield that gives you hyperarithmetic sets? | |
| Dec 15, 2014 at 17:18 | comment | added | Andreas Blass | By "a more refined absoluteness argument", do you mean getting absoluteness from some forcing model down to a model in which all subsets of $\omega$ are c.e.? That would seem to be too optimistic since the small model wouldn't be closed under taking complements of subsets of $\omega$. Another way to say that is that it's hard to imagine a Cohen real as looking like a c.e. set in any useful way. | |
| Dec 15, 2014 at 17:13 | comment | added | Joel David Hamkins | I had intended to inquire about whether we might undertake a more refined absoluteness argument, using actually generic objects, in order to produce a c.e. example of something, rather than merely making a priority argument look like forcing. But perhaps these aren't so different actually... | |
| Dec 15, 2014 at 16:55 | comment | added | Andreas Blass | On the one hand, I don't ordinarily say "priority" when all the requirements are compatible, so no prioritization is needed. On the other hand, I suppose the hierarchy "finite injury, infinite injury, monster, ..." could be extended backward to "0 injury", which would be Kleene-Post. As for making "genuine" priority arguments look like forcing, I think there have been many attempts over the years, but I haven't really learned about any of them. Some (surely not all) of the relevant names are Yates, Lachlan, and Lerman. | |
| Dec 15, 2014 at 15:58 | comment | added | Joel David Hamkins | Yes, I agree with that, although don't we think of the priority argument as a more refined and careful version of the simpler constructions? It is easier just to get the degrees somewhere, and the priority construction produces c.e. degrees. Do you know if one may use this forcing way of thinking to get actual c.e. degrees? | |
| Dec 15, 2014 at 15:41 | comment | added | Andreas Blass | Although you mentioned a priority construction, I think your last paragraph implies that priority isn't really involved. Won't "the construction of sufficiently generic degrees" amount to a Kleene-Post argument? (Priority would presumably become relevant if you wanted to do this with c.e. degrees.) | |
| Dec 15, 2014 at 12:07 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |