Re: Cardinality of Real Numbers

In <1125251641.615664.93900@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, on
08/28/2005
at 10:54 AM, jswimr3@xxxxxxxxx said:

>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.

That's not a mapping from N->R, it's a mapping from N->2^R.

>It seems like this would cover the full set of real numbers.

No, it wouldn't even cover all of the rational numbers.

>But it seems

Seems isn't good enough. Try to actually do it.

>like you could get around that by making a new rule,

No. But feel free to devise such 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.

That still covers only a subset of the rationals.

>So where did I go wrong?

By confusing your faulty intuition with a proof.

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