Timeline for Axiom of class collection
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 8, 2012 at 20:12 | answer | added | Stephan Alexander Spahn | timeline score: 2 | |
| Oct 16, 2011 at 0:23 | comment | added | Trevor Wilson | I don't have an answer, just a comment about the first paragraph of the question. Unless I'm mistaken the axiom stated there follows from global choice: take $C = B$, take the map $C \to B$ to be the identity, and take the map $C \to A$ to be a right inverse of the surjection $A \to B$. So it cannot imply the axiom of replacement as it is usually formulated. Should the map $C \to A$ be required to be a set? | |
| Oct 15, 2011 at 1:55 | comment | added | Daniel Mehkeri | Hmm, assuming my interpretation is right, then what happens for the specific case $\chi(x) :\leftrightarrow \exists y \phi(x,y)$ ? The existence hypothesis is trivial and we have no useful witnessing information on the LHS to work with, unlike in the set-collection case. | |
| Oct 15, 2011 at 1:35 | comment | added | Daniel Mehkeri | @Gerhard Paseman: Given classes $\chi,\phi$ such that $\forall x (\chi(x) \to \exists y \phi(x,y))$ then there is a class $\psi \subseteq \phi$ such that $\forall x (\chi(x) \to \exists y \psi(x,y))$ and $\forall x (\chi(x) \to \exists s \forall y (y \in s \leftrightarrow \psi(x,y)))$. Scott's trick is non-constructive, but (set-)collection is constructive, so he is proposing a class-collection axiom as a constructive substitute for Scott's trick. (@Mike Shulman: is that right?) Interesting proposal, no idea if it works. | |
| Oct 14, 2011 at 20:51 | comment | added | Mike Shulman | Michael Warren - no, I haven't, but that's a good suggestion. You're undoubtedly more familiar with such models than I am... (-: | |
| Oct 13, 2011 at 17:40 | comment | added | Gerhard Paseman | Also, an example pitched to an undergraduate level might help. Presenting actual statements of replacement, collection, and your version in slightly more formal terms might help, e.g. "Lets pick the language of NBG so we can talk about classes; I want to know more about the schema: for any first order formula phi(x,y) that has this property <insert surjection phrase>, there is a formula psi(x,y) that <insert surjection and factor phrase> and for which we assert <appropriate combination of phi and psi and b> is a set for all sets b." Gerhard "Ask Me About System Design" Paseman, 2011.10.13 | |
| Oct 13, 2011 at 17:31 | comment | added | Gerhard Paseman | If you are willing to cater to my specific ignorance, sure. I essentially want undergraduate-level explanations of fiber and small, something like "fiber is like the inverse image of a point; in relational terms, fiber(b) is like the class C of sets c such that the substitution phi(c,b) holds for an appropriate formula phi which might describe a mapping between classes." Similarly "In algebraic set theory, small means not large, so think of small as meaning no bigger than the size of some unfixed big set." Sloppy, but helpful. Gerhard "Coffee First, Then Set Theory" Paseman, 2011.10.13. | |
| Oct 13, 2011 at 13:26 | comment | added | Michael A Warren | Mike, I wonder if you have tried to prove that this holds in any known models of intuitionistic set theory? E.g., I would be tempted to look at these models in categories of ideals that have been considered in the algebraic set theory literature (such models certainly satisfy collection). | |
| Oct 12, 2011 at 16:28 | comment | added | Mike Shulman | Also, can you say what you don't understand about the question, so that I can attempt to clarify it? | |
| Oct 12, 2011 at 16:28 | comment | added | Mike Shulman | I haven't found any reference to such a thing in the anti-foundation literature that I've looked at. | |
| Oct 12, 2011 at 5:34 | comment | added | Gerhard Paseman | I am pretty sure I do not understand the question. However, I feel that "Scott's trick" (pick a representative of minimal rank out of each equivalence class) means that the answer is no for set theories with Foundation. You might look at those texts (Aczel, maybe?) whose authors work without Foundation. Your question may be of some interest there. Gerhard "Ask Me About System Design" Paseman, 2011.10.11 | |
| Oct 11, 2011 at 19:37 | history | asked | Mike Shulman | CC BY-SA 3.0 |