Re: 2 questions about AC

On Thu, 09 Jun 2005 01:47:02 GMT, "big.ass"
<big.ass.invalid@xxxxxxxxxxxx> wrote:

>Hello,
>AC for a finite set of sets is equivalent to the axiom of pair. In other
>words if we consider the set
>
>S={S_1, S_2, ... , S_k}
>
>with S_i<>{} for all i, we know that there are s_1 in S_1, s_2 in s_2 etc.
>such that (by axiom of pair):
>
>{(S_1,s_1),(s_2,s_2), ... , (S_k,s_k)};
>
>and this is a choice function.
>If S is not finite, AC is not equivalent to any other axiom of ZF. My first
>question is: extending axiom of pair to infinite is equivalent to accept AC?
>
>
>
>
>The second question is about the nature of the choice function. Considering
>Russel's socks example, we could comprehend that there is no acceptable law
>that allows to choose one sock.
>What is the "so inacceptable" thing with this method? At first glance, the
>function defined by choosing only one element of a class (despite of
>choosing "which" element is) seems not so inacceptable to me.

That's not a definition of a function.

>Thank you.
>


************************

David C. Ullrich
.

Relevant Pages

  • Re: Axiom of choice
    ... The difference between choice and determinacy, ... is, on the conception most, but not all, people seem to be compelled ... Doesn't a choice function imply the ability to serialize? ... view in favor of the full axiom of choice. ...
    (sci.math)
  • Re: Axiom of choice
    ... The difference between choice and determinacy, ... is, on the conception most, but not all, people seem to be compelled ... Doesn't a choice function imply the ability to serialize? ... view in favor of the full axiom of choice. ...
    (sci.math)
  • 2 questions about AC
    ... AC for a finite set of sets is equivalent to the axiom of pair. ... The second question is about the nature of the choice function. ... choosing "which" element is) seems not so inacceptable to me. ...
    (sci.logic)
  • Re: A poosible additional axiom to ZFC?
    ... >>that if a choice function c exists solely on the basis of AC, ... if f1, f2 are choice functions and _both_ exist solely on the basis of AC, ... axioms of ZFC would guarantee the xistence of a lot of sets from: ... new axiom, though, is that these choice functions do _not_ exist solely on ...
    (sci.logic)
  • Re: Dont get Axiom of Choice?
    ... family of non-empty sets on which we're trying to define a choice function. ... Continuing this, if S has only finitely many elements, then the pairing axiom ... If you can give an explicit rule for the choice function, ... and then the axiom of replacement ensures that ...
    (sci.math)