Re: Cardinality of Real Numbers


<jswimr3@xxxxxxxxx> wrote in message
news:1125251641.615664.93900@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> I've been thinking about cardinality proofs lately, and I've run into
> something that's been bothering me. I thought of what seems like a
> mapping from the set of integers to the set of real numbers. Now, of
> course, this can't exist, so there must be something wrong with my
> mapping, but I can't see what it is.
>
> The mapping works like this: for each integer, map it onto all the
> reals you can get by putting a decimal point anywhere in it. For
> example, 123 would map to:
>
> 123
> 12.3
> 1.23
> .123
>
> It seems like this would cover the full set of real numbers. Each of
> these mapped sets of reals is finite, and there would be a countable
> number of these sets, since the integers are countable. So this would
> seem to be a countable union of finite sets, which would, itself, be
> countable.
>
> I was wondering if perhaps I run into trouble with real numbers like
> .00000123, which wouldn't correspond to an integer in my scheme. But
> it seems like you could get around that by making a new rule, for
> example, that real numbers which begin with 1 would map to the numbers
> they would normally map to, but would also map to decimals where the 1
> is turned into a zero. So 10000123 would map to all the numbers it
> normally maps to, and would also map to .00000123. There would still
> be a finite number of real numbers for each integer.
>
> But the real numbers aren't countable. So where did I go wrong?

1. every number you "map to" is rational.
2. you do not even "map to" all of the rationals. (what about
..33333333333.... = 1/3? )


.

Relevant Pages

  • Re: Epistemology 201: The Science of Science
    ... This involves defining cardinality in terms of mapping. ... In the case of integers and rationals, we can map the integers into ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... This involves defining cardinality in terms of mapping. ... In the case of integers and rationals, we can map the integers into ...
    (sci.cognitive)
  • Re: Epistemology 201: The Science of Science
    ... This involves defining cardinality in terms of mapping. ... In the case of integers and rationals, we can map the integers into ...
    (sci.physics)
  • Re: learnable dynamics (was Goal of AI...)
    ... First, "lambda" is just the name, some arbitrary name I chose, for a real ... map the constant I called "k" was represented by the greek letter lambda. ... representations of reals. ... LogisticMap<> map; ...
    (comp.ai.philosophy)
  • Re: Epistemology 201: The Science of Science
    ... > In the case of integers and rationals, we can map the integers into ... significant way from ordering the sets to achieve your mapping. ... so this is not an example of comparing infinite sets without ...
    (sci.math)