# The Annotated Turing

## Chapter 2 - The Irrational and the Trascendental

There's one thing in chapter 2 (page 31) I find hardly convincing.

Petzold is talking about the Continuum Hypothesis, formulated (but not proven) by Cantor.

\[ \aleph_1 = 2^{\aleph_0} \]

Before that, he shows that \(|\mathbb{R}| = 2^{\aleph_0}\).

The general rule is that:

\[ \textrm{cardinality of a power set} = 2^{\textrm{cardinality of the original set}} \]

Here what happens if we construct the power set of the natural numbers.

We must find some order in our operations. The infinite sequence of natural numbers is set as a header. Then, for each row, we put 0s or 1s depending on whether the corresponding number is in the current element of the power set.

0 | 1 | 2 | 3 | 4 | 5 | … | Subset |
---|---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 0 | … | \(\{\}\) |

1 | 0 | 0 | 0 | 0 | 0 | … | \(\{0\}\) |

0 | 1 | 0 | 0 | 0 | 0 | … | \(\{1\}\) |

1 | 1 | 0 | 0 | 0 | 0 | … | \(\{0,1\}\) |

0 | 0 | 1 | 0 | 0 | 0 | … | \(\{2\}\) |

And so on. Proceeding like that we construct every possible infinite sequence of 0s and 1s, each one corresponding to one element of the powerset of \( \mathbb{N} \).

If these sequences of symbols are interpreted as binary encodings of
real numbers (like \( 100000… \rightarrow .100000… \)), then they are
**all** the real numbers between 0 and 1.

Since this set can be put in correspondence with the entire set of real numbers, it follows that the cardinality of \( \mathbb{R} \) and the cardinality of the powerset of \( \mathbb{N} \) are the same.

In a footnote, Petzold says that one must not believe that we found a way to enumerate all reals (which is indeed impossible), because there are no transcendental numbers in the set. That is true, Petzold says, because "each number has a finite number of 1s after the period".

I'm sure that is true, but didn't we just say that we listed **all**
real numbers? Isn't "all real numbers between 0 and 1 except for the
transcendental numbers" a different set?

This is what I find not very convincing.

**Update**

I posted a slightly modified version of this text as a question on math.stackexchange.