Timeline for Forcing for Arbitrary First Order Theories
Current License: CC BY-SA 3.0
10 events
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| Feb 19, 2014 at 23:26 | vote | accept | CommunityBot | ||
| Feb 15, 2014 at 13:59 | comment | added | Joel David Hamkins | One can show that every Boolean ultrapower is a direct limit of power set ultrapowers, using the directed system indexed by the maximal antichains in the Boolean algebra under refinement. | |
| Feb 15, 2014 at 2:05 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| Feb 15, 2014 at 1:10 | comment | added | Joel David Hamkins | Well, that is exactly what my paper with Dan Seabold is about: Well-founded Boolean ultrapowers as large cardinal embeddings jdh.hamkins.org/boolean-ultrapowers. | |
| Feb 15, 2014 at 0:58 | comment | added | user45939 | Thus we can interpret set theoretic forcing as a kind of ultraproduct. So the case seems a bit strange because using large cardinal assumptions one can form another form of ultraproducts of the universe too. Is there any direct relevance between Boolean valued forcing ultraproducts and large cardinal ultraproducts of the universe here? | |
| Feb 15, 2014 at 0:47 | comment | added | user45939 | Oh! I think I used the word "special" in an inappropriate way here. I didn't mean special models in the model theoretic sense. I meant a model with a special property as same as set theory that we use forcing to produce models of ZF with special properties. | |
| Feb 15, 2014 at 0:42 | comment | added | user45939 | The idea of interpreting forcing in Boolean valued version in order to generalize it to an arbitrary first order theory is very interesting. Did you construct any special model of a first order theory using this method before? By "special" I mean something like a special group, graph, field, etc. For example can we think about producing counterexamples using this kind of forcing for essential problems like finding a field which refutes Zilber's tricotomy conjecture in a different way from Hrushovski construction, etc? | |
| Feb 15, 2014 at 0:42 | comment | added | Joel David Hamkins | One should think of the Boolean ultrapower construction as a generalization of the ultrapower construction, which might be thought of as an averaging method, a method of producing homogeneous non-special models, rather than special models. | |
| Feb 15, 2014 at 0:27 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| Feb 14, 2014 at 23:36 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |