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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 ?

edited Jan 4 at 20:47

Asaf Karagila
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asked Jan 4 at 20:13

Rian
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What do you need this information for? Do you have a specific theorem that follows from ZFC but not from ZF+Countable choice? – Goldstern Jan 4 at 20:46

Goldstern · Accepted Answer · 2013-01-04 20:47:31Z UTC

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Dependent choice, for example. Or choice for well-ordered families, see http://mathoverflow.net/questions/118060

answered Jan 4 at 20:47

Goldstern
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Asaf Karagila · Answer 2 · 2013-01-04 20:47:24Z UTC

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:

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

what axioms are between AC and Countable choice !

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