Re: Cardinality of Real Numbers

>> I'm talking about an extension of Cantor's first, where
>> for any well-ordered set that bijects to the reals, there
>> are that many disjoint intervals generated.

Hi Ross, I'm having trouble with this first sentence. I need to break
it down so you can tell me wht you mean.

1. By "Cantor's first", do you mean Cantor's Theorem that the power set
of any set has strictly larger cardinality than the original set? If
so, I am not sure how that relates to the rest of the sentence.

2. What extension are you talking about? Is this his work or yours?

3. The term "well-ordered set" is redundant if we are talking about
ZFC. So are you assuming the Axiom of Choice or not?

4. The clause "for any well-ordered set that bijects to the reals" is
equivalent to "any set with cardinality C" (in ZFC). Is this what you
mean?

5. The line "there are that many disjoint intervals generated" is
fague. What many? It's certainly not C, since you can have only
countably many disjoint intervals on the real line. Proof: Every
interval on the real line must contain a rational number. Since
disjoint intervals must contain separate rationals, there cannot be any
more intervals than there are rationals. Since the rationals are
countable, so then must be any set of disjoint intervals.

Jonathan Hoyle
Eastman Kodak

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