Timeline for Formal Definition of Finite Conditions
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 28, 2016 at 21:27 | comment | added | Asaf Karagila♦ | That's a reasonable answer (although admittedly a bit of a cop out). Personally, I'd say that these forcings are with finite conditions, maybe because you can't really have any proper forcing changing cofinalities (without collapsing), so you cannot have something with "countable conditions" which to loosely imply some kind of closure. | |
| Dec 28, 2016 at 13:23 | comment | added | Joel David Hamkins | Well, since as I said the term is used loosely, I think there is nothing at stake in the answer. But I believe that Radin forcing should count as finite conditions if and only if Prikry forcing counts. I'm inclined to say that Prikry forcing does not count, but I can see how someone would say that it does. | |
| Dec 28, 2016 at 8:44 | comment | added | Asaf Karagila♦ | My question to Sean in the comments applies here. Would you say that supercompact Radin forcing is a forcing with finite conditions? | |
| Dec 27, 2016 at 23:12 | comment | added | Joel David Hamkins | The finite conditions notion that I mentioned is equivalent to saying that the forcing notion embeds into the Levy collapse poset for some large enough cardinal, since every collection of finite partial functions can be seen as a suborder of that partial order. (But note that this embedding is not a complete embedding and will not respect forcing.) | |
| Dec 27, 2016 at 22:56 | comment | added | Andrej Bauer | That's interesting, since almost every notion of finiteness and compactness I am aware of fall under the order-theoretic notion. It might be worthwhile drilling into finiteness-in-forcing a bit more, ot see if there's anything else there. | |
| Dec 27, 2016 at 22:42 | comment | added | Joel David Hamkins | About finding an order-theoretic characterization, I admire your proposal and attempt to do this. I am less confident, however, that it will be useful, since I think the characterization that you get will not be invariant under equivalence-of-forcing. For example, we should not expect to be able to pass the property to the Boolean completions (since those elements have infinitely much information). | |
| Dec 27, 2016 at 22:40 | comment | added | Joel David Hamkins | In the finite-function case, it is often important since it generally makes the forcing notion amenable to the $\Delta$-system theorem, which can be used to prove a good chain condition. Perhaps this is the main significance. | |
| Dec 27, 2016 at 22:39 | comment | added | Andrej Bauer | Is there any mathematical significance to having finite forcing conditions? And assuming there is, shouldn't then there be a purely order-theoretic way of characterizing finite conditions? | |
| Dec 27, 2016 at 22:28 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
added 4 characters in body
|
| Dec 27, 2016 at 22:23 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |