Re: Epistemology 201: The Science of Science

From: Wolf Kirchmeir (wwolfkir_at_sympatico.ca)
Date: 02/07/05 

Date: Mon, 07 Feb 2005 00:35:10 -0500

Albert wrote:
[...]
>
> Oh, I see. Mathematical jargon. I was hoping for you to tell me how to
> count an infinite number.

Consider the difference between the set of integers and the set of
reals. Both are infinite sets. Integers can be counted. That is, no
matter what integer you propose, you can count exactly one more: Given
N, you can specify that the next integer N' = N + 1 for any N. So the
set is countable. (This is not a rigorous explanation, but it will have
to do. It depends on the notion that counting is a step-by-step listing
of the elements of a set.)

You can't do this with real numbers, since you can't specify what the
next real number in the sequence will be: Given R, what is R'? Given any
R and R', there will always be an R" such that R < R" <R'. Or, no matter
how arbitrarily small you make the difference E between R and R', there
will a number R" such (R' - R") < E and (R" - R) < E. So, the set of
reals is uncountable.

See?

Relevant Pages

  • Re: Calculus XOR Probability
    ... If a quantitative set is mapped in ascending order from the naturals, ... number of reals on the line. ... to the subsequent logic that claims such a set cannot have infinite values. ... standard orderings, since sets in general don't come with little tags ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... If a quantitative set is mapped in ascending order from the naturals, with each increment in the domain, the range increases by some amount. ... Like it's the number of unit intervals, and the number of reals in the unit interval. ... You are using a form of infinite induction, making a claim for an infinite set based on all finite initial segments of it. ... don't have a definition for an arbitrary set of its "standard ordering" ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... If a quantitative set is mapped in ascending order from the naturals, with each increment in the domain, the range increases by some amount. ... you had said that the existence ... Like it's the number of unit intervals, and the number of reals in the unit interval. ... You are using a form of infinite induction, making a claim for an infinite set based on all finite initial segments of it. ...
    (sci.math)
  • Re: Dial 999 for the real number line
    ... length n as initial strings followed by the expansion of pi after the nth ... long decimal expansion followed by an infinite expansion. ... part can be as long as we like, we have an infinite sequence of initial ... dense in the reals. ...
    (sci.math)
  • Re: Dial 999 for the real number line
    ... Keep in mind that whatever system of axioms you come up with will ... proofs of uncountablility of the reals, but that is not going to happen. ... I have a hang-up, it's about infinite sequences. ... My point is that mathematical existence has nothing to do with physical ...
    (sci.math)
Quantcast