Editing Cardinality of the continuum (section)

===Alternative explanation for {{not a typo|&cfr; {{=}} 2<sup>א<sub>&lrm;0</sub></sup>}}===
Every real number has at least one infinite [[decimal expansion]]. For example,
{{block indent|1=1/2 = 0.50000...}}
{{block indent|1=1/3 = 0.33333...}}
{{block indent|1=π = 3.14159....}}
(This is true even in the case the expansion repeats, as in the first two examples.)

In any given case, the number of decimal places is [[countable set|countable]] since they can be put into a [[one-to-one correspondence]] with the set of natural numbers <math>\mathbb{N}</math>. This makes it sensible to talk about, say, the first, the one-hundredth, or the millionth decimal place of π. Since the natural numbers have cardinality <math>\aleph_0,</math> each real number has <math>\aleph_0</math> digits in its expansion.

Since each real number can be broken into an integer part and a decimal fraction, we get:
{{block indent|<math>{\mathfrak c} \leq \aleph_0 \cdot 10^{\aleph_0} \leq 2^{\aleph_0} \cdot {(2^4)}^{\aleph_0} = 2^{\aleph_0 + 4 \cdot \aleph_0} = 2^{\aleph_0} </math>}}

where we used the fact that

{{block indent|<math>\aleph_0 + 4 \cdot \aleph_0 = \aleph_0 \,</math>}}

On the other hand, if we map <math>2 = \{0, 1\}</math> to <math>\{3, 7\}</math> and consider that decimal fractions containing only 3 or 7 are only a part of the real numbers, then we get

{{block indent|<math>2^{\aleph_0} \leq {\mathfrak c} \,</math>}}

and thus
{{block indent|<math>{\mathfrak c} = 2^{\aleph_0} \,.</math>}}