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Re: Re: What is zero divided by zero anyway?

by arturo (Vicar)
on Oct 08, 2002 at 23:49 UTC ( #203798=note: print w/replies, xml ) Need Help??


in reply to Re: What is zero divided by zero anyway?
in thread What is zero divided by zero anyway?

Not only that, but "infinity" doesn't name one number. There are different infinities ...

For example, the set of natural numbers (0,1,2,3,...), aka N, is a *denumerable* infinity ('denumerable' meaning, 'can be put in a 1:1 correspondence with the members of N' -- so N satisfies this definition trivially), as is the set of multiples of 10 -- EVEN THOUGH that set is a subset of N. The set of multiples of 10 is, intuitively, a tenth the size of N? But no, for every member of the set of multiples of 10, you can find a partner in N.

Moreover, there are non-denumerable infinities (memory gets fuzzy here): the power set (set of all subsets) of N is non-denumerable; and the set of real numbers is also non-denumerable -- there are too many points on the real line to put in a 1:1 correspondence with N (intuitively: though with the case of multiples of 10, there is a way to figure out what the next number is, there's no way to find the "next point" given a specific point on the real line).

Isn't math fun?

If not P, what? Q maybe?
"Sidney Morgenbesser"

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Re^3: What is zero divided by zero anyway?
by Aristotle (Chancellor) on Oct 09, 2002 at 17:03 UTC

    Yes, and it gets weirder than that if you think further along those lines. In sets beyond the natural numbers, among the real numbers f.ex, any two numbers are separated by an infinite set of numbers. In fact, because you can translate any set of reals delimited by two numbers (exclusive) to any other set of reals delimited by any two numbers (exclusive) with a single multiplication, this means that the distance between any two numbers is equally large.

    F.ex, the set of numbers between 1.0 and 2.0 exclusive is inifinite. No matter how many members you assume, there are still more numbers between those. The same is true for the set of numbers between 2.0 and 10.0 - it is infinite. But although the interval from 1.0 to 2.0 exclusive is of length 1, and that from 2.0 to 10.0 exclusive is of length 8, you can map the set { x ∈ R ; 1.0 < x < 2.0 } to set { x ∈ R ; 2.0 < x < 10.0 } by simply multiplying by 8.

    As I said on the onset of this thread: zero and infinity are deep voodoo. Zero is deceptively so, it looks innocent at first, but it's still voodoo.

    Makeshifts last the longest.

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