*This question is quite soft, and I apologize in advance if it borderline off-topic.*

When working in theories between ZF and ZFC the term "choice principle" is heard quite often. For example:

$\quad$ **The Axiom of Countable Choice**: Every countable family of non-empty sets has a choice function.

We can say, if so, that a choice principle is an assertion about choosing. On the other hand, the above axiom is equivalent [1] to the following assertion:

$\quad$ Every $\sigma$-compact space is Lindelöf.

However this statement does not talk about choosing anything. The first alternative is to define a choice principle as something equivalent to a choice-referring statement, but what about the ordering principle?

$\quad$ **Ordering Principle**: Every set can be linearly ordered.

Commonly thought of as a choice principle, the order principle is not really equivalent to any choice principle directly. It does however prove that every family of non-empty and finite sets has a choice function.

Maybe we can define a choice principle as something which *proves* some sort of a choice statement. Alas this too can be troublesome, consider the Small Violations of Choice principle:

$\quad$ **SVC**: There exists $S$ such that for every $x$ there is an ordinal $\alpha$ such that $x\leq\mathcal P(\alpha)\times S$.

It is consistent with the existence of an amorphous set AND a countable set of pairs without a choice function that SVC holds, so it doesn't even imply the axiom of choice for pairs.

So we may wish to define a choice principle as follows:

Let $\varphi$ be a sentence in the language of set theory, $\varphi$ is a choice principle if ZF does not prove $\varphi$, but ZFC does.

This definition catches the wanted properties of a choice principle, it is simply an assertion in between the two theories. However in a recent comment on math.SE Carl Mummert suggested that this definition is also problematic, since the assertion: $$\lnot\rm AC\rightarrow\text{Con}(ZF)$$ Is not provable from ZF, but vacuously true in ZFC.

This leads to the question:

Question:What would be a good definition forChoice Principle(in ZF), which encapsulates the notion of a statement "between" ZF and ZFC, but still avoids vacuous statements as above?

**Bibliography:**

- Brunner, N.
**σ-kompakte Räume. (German. English summary) [σ-compact spaces]***Manuscripta Math.***38**(1982), no. 3, 375–379.

KWP($1$)(Every set can be embedded into the power set of an ordinal); how would you describe the general principle, namelyKWP($n$)"Every set can be embedded into the $n$-th power set iteration of some ordinal"? $\endgroup$ – Asaf Karagila Aug 5 '12 at 21:42