Timeline for Is there a universal way to force the Axiom of Choice to be true?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 28, 2018 at 21:47 | comment | added | Oscar Cunningham | Haha, okay. So I do still need some more conditions if I want the construction to be natural in any sense. | |
| Mar 28, 2018 at 21:26 | comment | added | Joel David Hamkins | Yes, there is such an interpretation: if AC holds, take all of V, otherwise go to L. | |
| Mar 28, 2018 at 13:16 | comment | added | Oscar Cunningham | But it would be nice if this was true whenever out original model obeyed choice. Do you know of an interpretation of ZFC in ZF that does this? i.e. Is there an interpretation of ZFC in ZF such that whenever you apply it to a model of ZF that happens to obey Choice the induced model is definably isomorphic to the original? That question is probably what I would have asked if I was trained to think in terms of logic rather than category theory. | |
| Mar 28, 2018 at 13:07 | comment | added | Oscar Cunningham | Ignore my above comment, I hadn't drunk enough coffee yet (I was thinking that bi-interpretability was a relationship between models, whereas in fact it's a relationship between theories). Am I right in thinking that ZF has an obvious interpretation in ZFC (just take the whole model)? So what we're looking for is an interpretation of ZFC in ZF with some nice properties. In this case (where one interpretation is trivial) the definition of bi-interpretation says that every model of ZF should be isomorphic to the model of ZFC it induces. Clearly this is stronger than what we want. | |
| Mar 28, 2018 at 10:21 | comment | added | Joel David Hamkins | I think I understood that. My point was that if you take models of set theory seriously as foundational realms, then you'd want the image model of a model to be accessible to the original model, in the way that the bi-interpretation insists upon. Otherwise the applications of your adjunction live in a place where no model can see it. | |
| Mar 28, 2018 at 8:01 | comment | added | Oscar Cunningham | Luckily I don't think we need a bi-interpretation. We're not looking for an adjunction between models of the set theory; we're looking for an adjunction between the categories of the models. This adjunction will induce a map from each model of ZF to the corresponding model of ZFC (or in the other direction depending on which way around the adjunction is). So we only need a morphism going one way rather than both. The category of models and interpretations looks like an interesting place to look for the adjunction in. | |
| Mar 27, 2018 at 21:31 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |