Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I’m not entirely happy with the introduction (“Statement”) to the page axiom of choice. On the one hand, it implies that the axiom of choice is something to be considered relative to a given category $C$ (which is reasonable), but it then proceeds to give the external formulation of AC for such a $C$, which I think is usually not the best meaning of “AC relative to $C$”. I would prefer to give the Statement as “every surjection in the category of sets splits” and then discuss later that analogous statements for other categories (including both internal and external ones) can also be called “axioms of choice” — but with emphasis on the internal ones, since they are what correspond to the original axiom of choice (for sets) in the internal logic.
(I would also prefer to change “epimorphism” for “surjection” or “regular/effective epimorphism”, especially when generalizing away from sets.)
I agree.
Since no one objected, I went ahead and made this change.
Seems like a somewhat roundabout way of putting it: can’t we just say that for infinite $X$, that $X$ and $F(X)$ have the same cardinality? Am I missing something?
Okay, I guess never mind my question. The direction that the indicated statement plus ZF implies AC doesn’t look easy.
Some of the equivalent statements to AC listed like
are suspicious as they talk about classes rather than sets. Isn’t it that AC for classes is a stronger statement ? Should one just put modifier small ?
Added equivalent statement
That any cartesian product of any family of inhabited sets is inhabited.
And it’s type theoretic analogue in the section on type theory
That any dependent product of any family of pointed sets is pointed.
Anonymous
Re #10:
These two clauses (here and here) were added in rev 75 by Anonymous and in rev 48 by Mike, respectively.
It seems clear that they are meant to be applied to small categories. For the clause mentioning strict categories this is almost explicit, since the entry strict category speaks as if smallness is the default assumption.
I have now added the smallness qualifier to both items, and also the strictness qualifier to the former.
adding reference
Anonymous
In May, an Anonymous editor added the following as an equivalent of AC:
That every subset $A \subseteq B$ in a universe $\mathcal{U}$ comes with a choice of injection $i:A \hookrightarrow B$. Constructive mathematicians usually use subsets equipped with the structure of an injection, as those are usually more useful than general subsets with the mere property of being a subset.
I don’t know what this means. How can a subset fail to come with a choice of injection defined by $i(a)=a$?
Perhaps by “subset $A\subseteq B$” it is meant a pair $A$, $B$ such that $|A|\le |B|$, or I guess that there merely exists an injection, and this is quantified over all pairs satisfying this cardinality condition. This feels like a Choice principle, but perhaps too strong?
I suppose that’s possible. That should follow from (global) choice, since you are choosing a set of specific injections from a family of nonempty sets of injections, and it certainly implies choice taking $A=1$.
Yes, it felt like Global Choice to me, which is stronger than the passage claims (plain AC). So, regardless, it will need editing a bit more.
Perhaps the relativization to “a universe” was intended to weaken it from global choice to ordinary choice?
1 to 21 of 21