Re: Epistemology 201: The Science of Science

stephen_at_nomail.com
Date: 02/10/05 

Date: 10 Feb 2005 21:24:21 GMT

In sci.math Tony Orlow (aeo6) <aeo6@cornell.edu> wrote:
: stephen@nomail.com said:
:> In sci.math Tony Orlow (aeo6) <aeo6@cornell.edu> wrote:
:> : stephen@nomail.com said:
:> :> In sci.math Tony Orlow (aeo6) <aeo6@cornell.edu> wrote:
:> :> : I never said that just because a result is counterintuitive that makes
:> :> : it wrong. There are accepted implications of Cantor's cardinalities that
:> :> : strike me as not just surprising and counterintutive, but contradictory
:> :> : to reality. This goes back to the need for mathematics, like other
:> :> : sciences, to test its predictions against reality.
:> :>
:> :> How do you propose to test the size of infinite sets against
:> :> reality? There are no infinite sets in reality as far as we know,
:> :> and there is no way we could compare infinite sets in our finite
:> :> lifetimes.
:> : (sigh) There is always the infinite set of points contained in a finite
:> : line segment, but that's a whole other topic.
:>
:> : No, wait! That's what I've been talking about for a week now! Now, how
:> : could I possibly explain how the number of points in a line segment,
:> : contained in another line segment, could possibly pose a problem for
:> : Cantorian cardinality? Hmmmm......... (yawn)
:>
:> You have yet to explain what you mean by the number of points in
:> a line. Just think about the phrase "number of points". What
:> is this number that you are using to describe the quantity of
:> points on a line? What rules does this number obey?
:> You refuse to define this number, and claim that it must
:> obey certain undefined rules. You have not explained anything,
:> and until you define your terms, you never will.
:>
:> You cannot deny that there exists a one-to-one correspondence
:> between the points in [0,1] and the points in [0,2]. That
:> is what Cantorian cardinality tells us. Where is the problem?
:>
:> Stephen
:>
:>
: No matter how one populates the line segment, it can contain an infinite
: number of points. Let's say we bisect the segment, then each of its
: halves, then each of those, etc. We end up with 2^infinity subsegments.
: That's really irrelevant though.

The following does not define the "numbers" that you are objecting to.
It also contradicts your earlier assertion that infinity+1=infinity.

: If segment A is a proper subsegment of segment B, then the set of points
: in A is a proper subset of the set of points in B, much like the odd
: integers are a proper subset of the integers. The superset obviously
: contains all members that the subset includes, plus additional ones. By
: any definition of a number, except for Cantor's infinity, the sum of a
: number and a non-zero number cannot be that original number. Cardinality
: ignores this fact to the extent that people tell me there are no more
: points in the segment than in the subsegment, which is patently insane,
: when you can point to and identify the additonal points. I can't
: understand how people can suspend their senses so much that this doesn't
: bother them. But, then, people live with a lot of contradictions that
: make no sense to me.

You apparently accept the contradiction that infinity+1=infinity and
that
  "the sum of a : number and a non-zero number cannot be that original number."
Of course you can say that infinity is not a number, but then you should
not say that there are an infinite number of points in a line. You
cannot have it both ways.

Again you are not appealing to logic. What is "patently insane" about
the fact that infinite cardinalities do not behave live finite ones?
If they behaved exactly like finite ones why would they not also be finite?

Stephen

Relevant Pages

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    (sci.math)
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