"Axiom of global choice" - MathOverflow

"Axiom of global choice"

2012-09-20T05:04:15Z

In some books on category theory (for example, in J.Adámek, H.Herrlich, E.Strecker "Abstract and concrete categories...") the authors use the idea of "big sets" ("conglomerates" or "collections") which can contain classes (as far as I understand, in the Goedel-Bernays sense) as elements, and they formulate the "generalized axiom of choice", where it is stated that the choice function exists (not only for families or classes of sets, but also) for families of classes (indexed by elements of those "big sets"). This approach allows to prove, in particular, the existence of a skeleton in each category, and some other useful things.

This generalization of the axiom of choice is also mentioned In Wikipedia: http://en.wikipedia.org/wiki/Axiom_of_global_choice (as the "strong form of the axiom of global choice").

I wonder if there are any texts with the justification of this trick? The references I found (in particular, those mentioned in Wikipedia) give justification only for usual axiom of choice (for families of sets or for classes of sets, but not for "conglomerates of classes"). So actually I can't understand whether, for example, the existence of a skeleton, is true for all categories (in some interpretation of set theory) or for some special ones... Similarly the other corollaries of this "global axiom of choice" look doubtful for me. I would be grateful if anybody could clarify this.

UPDATE 21.09.2012

From the comments I see that there is a risk of misunderstanding, so I want to explain that by justification I mean an accurate (rigorous) definition of the new tool together with the analysis of whether it is compatible with the other tools of the theory.

As an illustration, in the case of the usual axiom of choice (I mean its "weak form", in terms of Wikipedia), there are many textbooks (I can recommend E.Mendelson "Introduction to mathematical logic" or J.Kelly "General topology", the appendix), where the fundamental objects of the theory (in this case, the classes) are accurately introduced (here, axiomatically) and the necessary constructions (like functions) are rigorously defined in the theory. This makes possible to give rigorous formulation to the axiom of choice (again, to its "weak form") inside the theory, and moreover, this presentation of a new axiom is followed by a thorough investigation of whether it contradicts to the previous axioms of the theory. Only after receiving the answer that no contradictions can appear (in fact, a more strong thing is true: the new axiom is independent from the others, that was the result by P.Cohen) mathematicians can use this new axiom without worrying that something is wrong here.

So my question is whether there is something similar for the "strong form of the axiom of choice"? Is it possible that nothing lies behind these words? On the contrary, if there is a justification, where can I read about it?

UPDATE 21.04.2013

Dear colleagues, from what I learn on this subject in the textbooks which I found, in Wikipedia and here in MO, I deduce that what people call "axiom of global choice" is just the usual axiom of choice (as it is presented in Kelly's book) applied to some special classes of sets arising in consideration of what is called the Grothendieck Universe. It's a puzzle for me

1) why people call this special case "a stronger form of the axiom of choice", and

2) why they don't want to give references, where this construction is accurately introduced.

With the aim to accelerate the clarification of this question, I now nominated for deletion the article in Wikipedia devoted to his topic: http://en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion/Axiom_of_global_choice. As I wrote there, I don't exclude that the partisans of the idea will rewrite the article in Wikipedia for endowing "global choice" with some mathematical sense, but you should agree that in its present form this article and the other mentionings of "global choice" available for external observers, look indecently vague. I invite all comers to share their opinion here or at the Wikipedia page.

Answer by Dimitri Zaganidis for "Axiom of global choice"

2013-03-21T11:50:56Z

I think that you should place yourself in ZFC+ existence of strongly innaccessible cardinals.

Then the existence of a strongly inaccessible cardinal provides you a universe as in Borceux's Handbook of Categorical algebra.

Then, what you call sets are elements of the universe, and what you call classes are the subsets of the universe, but they are still sets in the set theoretic sense, so you can apply choice.

EDIT: clarification

The problem of category theory is that we want to have the category Set of all sets to actually be a category.

Since there is no set which contains all sets, we can't ask a category to have a set of objects, or Set will no more be a category. That's why in the first place we define the collection of object to be potentially wider than a set: we ask it to be a class.

The point is that you can avoid the difficulty differently, by limiting yourself to a rich enough set of sets, which should contains "everthing that you can be interested in".

This is the concept of a Grothendieck universe, see http://ncatlab.org/nlab/show/Grothendieck+universe or Borceux's Handbook of Categorical algebra for a definition.

Existence of Grothendieck universes turns out to be equivalent to the existence of strongly inaccessible cardinals (here we are in ZFC), and this existence axiom has been studied in set theory (I'm not a specialist of that at all).

So you place yourself in ZFC + existence of strongly inaccessible cardinals, and you take a universe $U$. Call the elements of U the "U-sets" and the subsets of U the "U-classes".

Then, define a category $\mathscr C$ (in the universe $U$) to be a triple $(Obj~ \mathscr C, Mor~ \mathscr C, \circ)$ with $Obj~ \mathscr C \subseteq U$ and $\mathscr C(A,B) \in U$ for all $A,B \in Obj~ \mathscr C$, satisfying the usual axioms of a category.

So now, your U-classes are indeed sets of ZFC (as subsets of the set U), so you can use the axiom of choice in them, without bothering.

I am not sure if it is what you were looking for, but it is what I personally have in mind when I am using the axiom of choice to choose in a collection of objects, in category theory.

Answer by Joel David Hamkins for "Axiom of global choice"

2013-04-13T16:00:24Z

Here at least is the usual justification for moving from AC for sets to what is normally called the global axiom of choice, which asserts that there is a class well-ordering of the (first-order) universe.

Theorem.

The global axiom of choice, when added to the ZFC or GB+AC axioms of set theory, leads to no new theorems about sets. That is, the first-order assertions about sets that are provable in GBC are precisely the same as the theorems of ZFC.
Furthermore, every model of ZFC can be extended (by forcing) to a model of GBC, in which the global axiom of choice is true, while adding no new sets (only classes).
In particular, the global axiom of choice is safe in the sense that it will not cause inconsistency, unless the underlying system without the global axiom of choice was already inconsistent.

Proof. Suppose that $M$ is any model of ZFC. Consider the class partial order $\mathbb{P}$ consisting of all well-orderings in $M$ of any set in $M$, ordered by end-extension. As a forcing notion, this partial order is $\kappa$-closed for every $\kappa$ in $M$, since the union of a chain of (end-extending) well-orderings is still a well-order. If $G\subset\mathbb{P}$ is $M$-generic for this partial order, then $G$ is, in effect, a well-ordering of all the sets in $M$. Furthermore, one can prove by the usual forcing technology that the structure $\langle M,{\in},G\rangle$ satisfies $\text{ZFC}(G)$, that is, where the predicate $G$ is allowed to appear in the replacement and other axiom schemes.

Essentially, what we've done is add a global well-ordering of the universe generically. And since the forcing was closed, no new sets were added, and so $M[G]$ has the same first-order part as $M$.

It follows now that GBC is conservative over ZFC for first-order assertions, since any first-order statement $\sigma$ that is true in all GBC models will be true in $M[G]$ and therefore also in $M$, and so $\sigma$ is true in all ZFC models as well. QED

Answer by Fred Rohrer for "Axiom of global choice"

2013-04-21T09:49:45Z

Dear Sergei, you might be interested in first reading Bourbaki's Théorie des ensembles (at least chapters I--III) and then have a look at section 0 and the appendix of SGA 4.I. This gives a slightly different approach using Hilbert's almighty symbol $\tau$.