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Hi All. Need some information. We all know axiom of choice (AC) and countable choice. Which axioms are between these two. I mean weaker that Axiom of choice but stronger than countable choice ?

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    $\begingroup$ What do you need this information for? Do you have a specific theorem that follows from ZFC but not from ZF+Countable choice? $\endgroup$
    – Goldstern
    Commented Jan 4, 2013 at 20:46

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Dependent choice, for example. Or choice for well-ordered families, see Existence of model of ZF without AC, but with many choice function

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There are many many many axioms.

For example you have the Principle of Dependent Choice, you also have the generalized axiom of choice for $\kappa$, "For every family of size $\kappa$ of non-empty sets there is a choice function." and whenever $\kappa>\aleph_0$ this is a strictly stronger axiom. In a similar fashion the principle of dependent choice is extended to what is known as $\mathsf{DC}_\kappa$.

There are many choice principles and they come in different flavours and shapes. Some good places to start would be:

  1. Howard & Rubin's Consequences of the Axiom of Choice.
  2. Herrlich's The Axiom of Choice.
  3. G. Moore's Zermelo's Axiom of Choice.
  4. Jech's The Axiom of Choice.
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