"The Annotated Turing" on listing all binary numbers between 0 and 1 In his book &quot;The Annotated Turing&quot; in the first sentence on page 32 Charles Petzold wrote:

These are binary numbers between 0 and 1, and (judging from the way we
created these numbers) all the binary numbers between 0 and 1, and
hence all the real numbers between 0 and 1.

And then at the bottom of the page there is a note:

It may also seem as if we've stumbled on a method to enumerate all the
real numbers between 0 and 1...

Followed by a brief explanation of why we really have not.
These two sentences contradict each other. The first one says we've made a list of all real numbers and the bottom note says we have not. So I have to questions regarding this:

Is it a mistake or am I getting something wrong?
If we have not listed all the real numbers between 0 and 1, then what have we listed?
You haven't shown the full proof, but it most likely has the form &quot;Assume that the set is countable.  Consider any enumeration.  Then (blah-blah-blah) shows that there is a number not in the enumeration.  That's a contradiction, so our assumption must have been wrong.&quot;
If that's the structure of the proof, then pointing out that the proof appears to say two contradictory things is not a surprise; that's entirely the point of the line of reasoning, to show that if you start with a particular assumption, you reach absurd results.
What's actually true?  The truth: that list doesn't actually contain all the real numbers.  Assuming that it does is a faulty assumption.
It's possible that the text could have been written a little more carefully to caveat that the statement about it containing all numbers; but oh well, no one's perfect.  Books written for a popular audience, in particular, aren't always the best source if you want a precise and careful proof of a mathematical theorem.