Re: infinity
- From: "Randy Poe" <poespam-trap@xxxxxxxxx>
- Date: 18 Oct 2005 12:14:36 -0700
Tony Orlow wrote:
> Virgil said:
> > Except that Card(P(N)) = Card(R) has been proved to hold outside of
> > TOmatics.
> Only according to axioms that basically state this fact with no justification.
It's a theorem. That means it comes with a proof.
> There is no justification for any such statement. If you disagree, please
> explain exactly why this is so.
1. The meaning of "Card(A) = Card(B)" is defined as "there exists
a bijection between A and B."
2. A bijection between P(N) and R can be shown to exist.
3. Therefore Card(P(N)) = Card(R)
Now, you may not like #1, but in your system there is no such
thing as "cardinality of an infinite set". Your cardinalities are
"unspecifiable numbers" with "tenuous existence".
In our axiomatic system, #1 is the meaning of "equal cardinality"
and #2 is the proof.
- Randy
.
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