Re: axiom of choice and equivalent statements
- From: "Martin" <sleziak@xxxxxxxxxxxxx>
- Date: 5 Feb 2006 10:49:07 -0800
Some of them you can find here:
http://planetmath.org/encyclopedia/AxiomOfChoice.html
or
http://darkwing.uoregon.edu/~ross1/SetTheory.pdf
Just few notes to the other forms:
(2) => (4) Let W be any set.
For any ordinal alpha either W<=alpha or alpha<=W holds.
It's not possible that alpha<=W for all ordinal numbers (leads to a
contradiction, W would be a proper class).
So we have an injection f:W->alpha. The relation
x <_W y <=> f(x) < f(y)
is the well-ordering of W.
(Maybe a less-formal statement is better for intuition: the injection f
realizes W as a subset of the ordinal alpha. Every subset of a
well-ordered set is well-ordered.)
To (5): This doesn't hold for an empty relation - something is missing
in the condition.
If R is such that to each x e dom R there exists at least one y with
xRy, the F is simply choice function for the system
A_x={y; xRy},
where x e X.
To show that AC implies (6) it might be helpful to note that elements
of product of non-empty sets X_i, i e I, are precisely the choice
functions I->X_i.
(Multiplicative form of AC is the statement that the product of
non-empty sets is non-empty.)
Best regards
M
.
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