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Science Forum Index » Logic Forum » The intuitive basis of set size
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| Guest |
 Posted: Sat May 03, 2008 3:27 pm |
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Hi all,
Lets say that Size 'S' is a primitive concept.
Assuming Choice.
let's add the following two axioms:
Axiom 1 : Sx=Sy -> Exists f : f is a bijection between x and y.
Axiom 2 : ~ Sx=Sy -> [Exists f : f is a strict injection from x to y
or Exists g : g is a strict injection
from y to x].
Lemma : suppose that x and y are finite, then it follows that:
[Exists f : f is a strict injection from x to y
or Exists g: g is a strict injection from y to x]
<->
~ Exists f: f is a bijection between x and y.
Accordingly:
~ Sx=Sy -> ~ Exists f : f is a bijection between x and y.
So
Exists f : f is a bijection between x and y -> Sx=Sy
Accordingly the following theorem is true:
Theorem 1 : for finite sets x and y:
Sx=Sy <-> Exists f : f is a bijection between x and y
But this is only for finites!
When matters come to infinites things do differ, since
for infinite sets x and y we DON'T have the Lemma above.
So Theorem 1 doesn't follow.
But we see that Cantor had DEFINED 'Cardinality' of sets, in such a
manner that :
Cardinality(x) = Cardinality(y)
<->
Exists f : f is a bijection between x and y
for ANY sets x and y.
Why would that be a justified definition?
Zuhair |
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