Re: Cardinality of Real Numbers

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.

Stephen
.

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