Timeline for Does this axiom (a weak form of class valued choice) has a name?
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 7, 2015 at 15:43 | vote | accept | Simon Henry | ||
| Aug 7, 2015 at 15:26 | answer | added | Joel David Hamkins | timeline score: 3 | |
| Aug 7, 2015 at 15:25 | comment | added | Simon Henry | @ZhenLin : thank you, I just check the book and this is indeed exactly the "collection axiom" when we write it in terms of categorical logic. | |
| Aug 7, 2015 at 14:29 | comment | added | Zhen Lin | I think this is a variation of (algebraic set theory version of) the axiom of collection. See [Joyal and Moerdijk, Algebraic set theory, Ch. I, §1]. | |
| Aug 7, 2015 at 13:52 | history | edited | Simon Henry | CC BY-SA 3.0 |
added 296 characters in body
|
| S Aug 7, 2015 at 13:45 | history | suggested | Tadashi |
Added relevant tag
|
|
| Aug 7, 2015 at 13:35 | review | Suggested edits | |||
| S Aug 7, 2015 at 13:45 | |||||
| Aug 7, 2015 at 12:23 | comment | added | Simon Henry | Sorry, maybe I shouldn't have use the word 'class' at all. I don't mean the 'meta' notion (a familly of sets that can be defined by a formula). I assume that I am working in a framework where I have a notion of class satisfying some axioms (so the question " what is a class" is basically the same thing as "what is a set" when working with ZF) and I don't really care on whether or not class can be element of class... In my mind this distinction was irrelevant for the question, But Eric Wofsey comments showed me that it is... | |
| Aug 7, 2015 at 12:08 | comment | added | Asaf Karagila♦ | I'm sorry, but I still don't fully understand what you mean by class. | |
| Aug 7, 2015 at 11:56 | comment | added | Simon Henry | Ok if you are assuming that proper class cannot be element of other class then you are ok. But I prefer to consider that my 'class' behave like sets (for exemple I want to be able to talk about the class of functions between two given class) which is not possible if class cannot be element of class. So I can use either ordinary set theory + one Grothendieck universe as I said in my previous comment, or take ordinary set theory (calling the object class instead of set) with one additional predicate that distinguishes 'set' from 'class' + a few additional axioms... | |
| Aug 7, 2015 at 11:56 | comment | added | Eric Wofsey | In the framework of ZF, this (stated as a theorem schema) follows from Foundation by Scott's trick (and does not involve Choice at all). | |
| Aug 7, 2015 at 11:46 | comment | added | Asaf Karagila♦ | Okay, so I need to ask what is a class? Because I know that a class is a collection of sets (or, if you prefer, a "subset" of the universe). Proper classes are never elements of other classes, and in fact that is a good way to distinguish sets from classes. Sets are classes which are elements of other classes. So if $A$ is a proper class, $\{A\}$ is not anything as far as theory goes, it is singleton in the meta-theory, sure. But if in the meta-theory your universe is countable, what does it mean to have a bijection with a set? So I'm a bit confused here. | |
| Aug 7, 2015 at 11:42 | comment | added | Simon Henry | @AsafKaragila: A small class is a class which is in bijection of a set. It might yields some problem to says that it is a set depending on your foundation: for example if it is ordinary set theory with one Grothendieck universe (where we say sets for element of the universe and class for arbitrary 'set' no necessary in the universe ), you will run into trouble if you ask that for every class $A$, the small class $\{A\}$ is an element of the universe, it will allow to construct the class of all class etc... | |
| Aug 7, 2015 at 10:47 | comment | added | Asaf Karagila♦ | What is a small subclass? Is it not a set? | |
| Aug 7, 2015 at 10:11 | history | asked | Simon Henry | CC BY-SA 3.0 |