On Fri, May 27, 2011 at 2:52 PM, Daniel C. <[email protected]> wrote:
> No. I meant to say exactly what I said - that there are in fact
> "more" numbers between 0-2 than there are between 0-1. That is, the
> cardinality of the set of all numbers between zero and one is of a
> lower order than the cardinality of the set of all numbers between
> zero and two. However, since neither of these sets are countable, the
> concept or "more" or "less" (which applies just fine to finite sets -
> iirc, all finite sets are countable) breaks down when talking about
> infinite sets, especially ones that are uncountable.
>
> I *believe* that I am using the phrase "the cardinality of (the set)
> is greater" correctly in the above paragraph, but it's been a few
> years; I trust that Levi will correct me if I'm using the words
> incorrectly, or if I'm just plain wrong =)
I'm afraid you're just plain wrong here. From the 'Cardinality'
article that you referenced:
Cardinal arithmetic can be used to show not only that the number of
points in a real number line is equal to the number of points in any
segment of that line, but that this is equal to the number of points
on a plane and, indeed, in any finite-dimensional space. These results
are highly counterintuitive, because they imply that there exist
proper subsets and proper supersets of an infinite set S that have the
same size as S, although S contains elements that do not belong to its
subsets, and the supersets of S contain elements that are not included
in it.
The solution to the problem of how an infinite universe can expand
lies in first figuring out what you mean by 'infinite universe' and
what you mean by 'expand'. There is indeed a solution to this, given
appropriate definitions, which can be found on the 'Cosmology
Tutorial' page I linked in a different thread.
--Levi
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