Stronger than ZF, weaker than ZFC

Question

Can you please name axiom system that is strictly weaker than ZFC and strictly stronger than ZF? (such as DC, AC$_\omega$)

I searched for it but i could only find these two. If there are statements equivalent to DC or AC$_\omega$ in ZF, please tell me or give me a link introducing those.. (I want some equivalent statements, since I don't know where these two can be used usefully.)

Answer 1

There are infinitely many assertions which are between ZF and ZFC, these are often called choice principles.

Slightly more formally (but naively, as Carl indicates in the comments), we say that a sentence in the language of set theory $\varphi$ is a choice principle if it is provable from ZFC but not from ZF, we say that $\varphi$ is a weak choice principle if ZF+$\varphi$ does not prove AC.

Examples for choice principles:

1. the axiom of choice;
2. the axiom of choice for countable families;
3. countable unions of countable sets are countable;
4. the axiom of choice for countable families of finite sets;
5. the axiom of choice for countable families of pairs.

The first one is not a weak choice principle, it is in fact the axiom of choice in full. It can be shown that each choice principle implies the following, and that all those implications are strict. The list can be expanded much further, even if we require it will remain "linear" (that for every two statements one of them implies the other).

However there are other choice principles, for example:

The ultrafilter lemma: Every filter can be extended to an ultrafilter.

This is a particularly important choice principle with many equivalents and uses throughout mathematics. For example, it is equivalent to the compactness theorem in logic; or to the Tychonoff theorem for Hausdorff compact spaces. This choice principle implies that every set can be linearly ordered, as well the Hahn-Banach theorem and in turn - the Banach-Tarski paradox. From this family of choice principles also follow the fact that every field has an algebraic closure, and that the closure is unique up to isomorphism.

There are many many other choice principles, some of them are equivalent to the ultrafilter lemma; others to the axiom of choice for families of some sort (countable; etc.); and other principles are just "out there" and are not equivalent to the variants of these two above, for example KWP($n$) and SVC are two important examples.

Specifically for DC, we have an interesting equivalent:

• The Baire category theorem for complete metric spaces.

As for the axiom of countable choice, it too has several equivalents and one of the interesting ones, for example:

• $\sigma$-compact spaces are Lindelof.

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To learn more about such principles you may find yourself interested in reading these books, all contain proofs, lists and diagrams of various choice principles and the known implications between them:

1. Herrlich, H. Axiom of Choice. Lecture Notes in Mathematics, Springer, 2006.

2. Jech, T. The Axiom of Choice. North-Holland (1973).

3. Howard, P. and Rubin, J.E. Consequences of the Axiom of Choice. American Mathematical Soc. (1998). Also see the online database for the book.

4. Moore, G. H. Zermelo's Axiom of Choice. Springer-Verlag (1982).