Re: Cardinality of Real Numbers


stephen@xxxxxxxxxx wrote:
> jswimr3@xxxxxxxxx wrote:
> > 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.
>
> What about 1/3? Or sqrt(2)? All integers have a terminating
> decimal representation. Not all reals have a terminating decimal
> representation.
>
> <snip>
>
> > But the real numbers aren't countable. So where did I go wrong?
>
> By assuming that all reals have a terminating decimal
> representation.

This seems to be a common mistake made by people who claim that [0,1)
is countable.

--- Christopher Heckman

.

Relevant Pages

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