Timeline for Does the class category of ZF-algebras satisfy the Multiverse axioms?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 24, 2012 at 12:52 | comment | added | user2529 | Joel, thank you for your answer. Please do not delete it. I meant to ask something closer to what your reply is saying, although Mike's reply is the "correct" answer to my question as I had phrased it. I am still hoping that categorial techniques can be used to study your Multiverse. The 2-category of topoi contain topoi which have intuitionistic character. However, your Multiverse axioms remain in a classical/boolean flavor. It seems to me that forcing is just like extensions in algebra. Structural morphisms (eg elementary embeddings) should play a larger role in studying the Multiverse. | |
| Jan 24, 2012 at 12:46 | comment | added | user2529 | Joel, I like your last remark! | |
| Jan 23, 2012 at 19:19 | comment | added | Joel David Hamkins | mbsq, to continue: in particular, my view is that technical developments analogous to forcing will eventually arise, by which we shall be able to construct new set-theoretic universes allowing us to modify arithmetic truth with the same ease and flexibility with which we may currently modify higher order truth via forcing; this situation will undermine the current widespread confidence that there is an absolute notion of the finite, just as our experience with forcing has undermined confidence in absolute set-theoretic truth. | |
| Jan 23, 2012 at 16:59 | comment | added | Mike Shulman | I agree that this answer should be kept, and not only just because it shows thought processes. While it doesn't exactly answer the question I think it answers something quite close to the question, and should be useful to anyone thinking about relating these theories. | |
| Jan 23, 2012 at 16:57 | comment | added | Joel David Hamkins | The motivation is that our concept of well-foundedness is explicitly dependent on the set-theoretic background---we know the concept is not absolute to different models of set theory---and so once one makes the move to abandon the pretence of an absolute background concept of set, there seems to be no foundation for any absolute concept of well-foundedness. This is true even for our concept of natural number, since all the categoricity proofs for this concept (e.g. Dedekind's proof that second-order Peano is categorical) are based on having a fixed background concept of set. | |
| Jan 23, 2012 at 16:42 | comment | added | Monroe Eskew | @Joel- Just curious, what is the motivation for the w.f. mirage axiom? | |
| Jan 23, 2012 at 16:03 | comment | added | Steven Gubkin | Keep the answer around! There is still a lot of valuable information here. As a general rule, I think that keeping all (nonspam) answers is a good idea. It gives a record of thoughts in process, which is a VERY rare thing to find in mathematics. Mathoverflow is one of the few places in the world where you can read the real time thought process of a mathematician, not a polished distillation of that process. | |
| Jan 23, 2012 at 12:15 | comment | added | Joel David Hamkins | Vote this comment up if I should delete my answer. | |
| Jan 23, 2012 at 11:49 | comment | added | Joel David Hamkins | Ah, it seems from your answer that the idea is closer to a categorification of second order set theory. So the presumption of my answer that we had a category of models of set theory is off base. | |
| Jan 23, 2012 at 6:59 | comment | added | Mike Shulman | I think your understanding is close, but not quite right: the objects of a category of classes are just classes, without the extra structure of a membership relation satisfying any axioms. But one can of course consider objects of a category of classes equipped with such structure, and then I think your answer applies to such things. | |
| Jan 23, 2012 at 4:58 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |