Pete said:

I do see that the list does not contain all numbers.

.1 - that's all one-bit numbers.

.01, .11 - that's all two-bit numbers.

.001, .101, .011, .111 - that's all three-bit numbers.

.0001, .1001, .0101, .1101, .0011, .1011, .0111, .1111 - that's all four-bit numbers.

Etc.

It's just counting from one to infinity and reversing the order of the bits. It contains all numbers (between zero and one), it is mappable to integers (with the pesky problem of finite fractions mapping to integers of finite length), and it's ordered in the same sequence as the mapped integers.

If your list is to contain all numbers, then your list must include numbers with infinite digits. Does it?

Yes. They're all at the bottom of the list. But you can't actually see them because the list is infinitely long. Then again I suppose you couldn't "see" one of them anyway because you could only see the portion of it transcribed in a string of digits of arbitrary finite length.

I think that it does not, in fact, give a one-to-one mapping between integers and fractions. I think that one-to-one mapping means that for each finite integer, you must be able to find a matching finite fraction, and for each finite fraction, you must be able to find a matching finite integer.

Yes, I understand that now.

This could be a problem - is there such a thing as an infinite integer? I think that integers might be finite by definition.

I'm not sure what the definition of an integer is. I'm still struggling through the Wikipedia definition of "surjective."

But I thought the definition hinged on not having a fractional part. A "cardinal number" as we say in linguistics. I just don't know how to say that in proper mathematical terms.

But all integers are fractions, therefore if all fractions are finite then all integers are also finite, right?

No, for the purpose of this discussion an integer has the the part to the right of the decimal (or binary) point equal zero and a fraction has the part to the left equal zero. Using standard definitions, though, an integer is indeed a fraction with the denominator equal one.

To sum up, I see the error of my ways. I stick by my guns and maintain that my series is an ordered list of all possible fractions if it is allowed to go to infinity. But I agree that it does not satisfy the proper definition of "countable."

The definition of "surjective" is still in question.