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At some point in my work (which has nothing to do with set theoretics foundation) I need to consider the following axiom:

For any set $X$, any class $V$ with a surjective map $f : V \twoheadrightarrow X$ there exists a small subclass $V' \subset V$ such that the restriction of $f$ to $V'$ is already surjective.

(The general framework I'm working in is intuitionist mathematics with a notion of class, and small for a class mean in bijection with a set.)

It can be considered as a weak form of the axiom of choice, as the class valued axiom of choice would be about finding a $V'$ such that the restriction of $f$ is bijective (and in particular $V'$ would be small)

What I want to know is if this axiom appear somewhere in the literature. and especially does it have a name ? As it been studied or used ?

Edit : Eric Wofsey pointed out in the comment that in ZF (with the regularity axioms) this 'axiom' is in fact a theorem. This unfortunately does not apply to the framework I had in mind, but highly suggest that this probably does not have a name or hasn't been studied as an axiom anywhere...

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  • $\begingroup$ What is a small subclass? Is it not a set? $\endgroup$
    – Asaf Karagila ♦
    Commented Aug 7, 2015 at 10:47
  • $\begingroup$ @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... $\endgroup$
    – Simon Henry
    Commented Aug 7, 2015 at 11:42
  • $\begingroup$ 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. $\endgroup$
    – Asaf Karagila ♦
    Commented Aug 7, 2015 at 11:46
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    $\begingroup$ 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). $\endgroup$
    – Eric Wofsey
    Commented Aug 7, 2015 at 11:56
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    $\begingroup$ 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]. $\endgroup$
    – Zhen Lin
    Commented Aug 7, 2015 at 14:29

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In weak set theories, using classical logic and interpreting "small subclass" as "set", this principle amounts to an alternative formulation of the collection axiom. For example, in Zermelo set theory or even much weaker theories, even without the power set axiom, this principle is equivalent to the collection axiom scheme.

Collection: If $X$ is a set and $\forall a\in X\exists b\ \varphi(a,b)$, then there is a set $Y$ such that $\forall a\in X\exists b\in Y\ \varphi(a,b)$.

One can replace the talk of a formula $\varphi(a,b)$ with membership in a class of pairs.

If we have collection, then suppose we have a class surjective function $f:V\to X$ as in your case. Since for every $a\in X\exists b\ f(b)=a$, by collection we can find a set $Y$ such that for every $a\in X\exists b\in Y\ f(b)=a$, and so $f\upharpoonright Y:Y\to X$ is already surjective, as desired.

Conversely, if your axiom holds and we have a formula $\varphi(a,b)$ for an instance of collection, so that $\forall a\in X\exists b\ \varphi(a,b)$, then let $f(a,b)=a$, provided that $\varphi(a,b)$, so that $f$ is surjective from the class $\{(a,b)\mid \varphi(a,b)\}$ to $X$. Under your axiom, there is a set $Y$ such that $f\upharpoonright Y:Y\to X$ is already surjective. In this case, the projection of $Y$ onto the second coordinate, that is, the set $B=\{b\mid \exists a\ (a,b)\in Y\}$ is a set, and the surjectivity of the restriction amounts to $\forall a\in X\exists b\in B\ \varphi(a,b)$, thus verifying the desired instance of collection.

This argument doesn't seem to need much at all in the background set theory, although I am not sure what the effects would be without classical logic.

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  • $\begingroup$ Thank you, together with Zhen lin comment on the "categorical version" of this axiom it gives me exactly what I needed, and it indeed apply largely independently of the background theory. $\endgroup$
    – Simon Henry
    Commented Aug 7, 2015 at 15:44

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