2
$\begingroup$

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 ?

$\endgroup$
1
  • 3
    $\begingroup$ 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
    Jan 4, 2013 at 20:46

2 Answers 2

2
$\begingroup$

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

$\endgroup$
5
$\begingroup$

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.
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.