Hurkyl,
This is exactly the beautiful thing in {_} content, for example:
.__. = Finite line = [__]
__ = Infinite line = (__)
.__ = Infinite line = [__)
|{.__.}| = 1
|{.____.}| = 1
|{.________.}| = 1
|{.__}| = 1
|{__}| = 1
Shortly speaking, __ is the essence of an invareant self similarity over scales.
So how do you reconsile this with the fact that, if I assume the real numbers are countable, I can prove the real numbers have a length less than 1/2?
Anyways, one of your major problems is that you seem to confuse the order of the quantifiers in the statements we make.
For example, consider these two statements:
For any (non-bald) person P you pick, I can choose a color C, such that person P's hair is color C.
I can choose a color C, such that for any (non-bald) person P you pick, that person P's hair is color C.
One of these claims is very easy, and one of these is impossible! I hope this demonstrates why the order of these operations is important.
Sometimes, doing things procedurally helps understanding. You fulfill each
quantifier one step at a time before moving onto the next one.
For instance, I claim this is possible:
Step 1: You choose any person.
Step 2: I choose a single color.
Fact: The person you chose has hair with the color I chose.
Proof: At step 2, I can look at the person you chose, and select his hair color.
I claim this is not always possible.
Step 1: I choose a single color.
Step 2: You choose any person
Fact: The person you chose has hair with the color I chose.
Proof: At step 2, you know what color I chose, and you can choose a person with a different hair color.
The problem at hand (enumerating the list of binary sequences)
There exists a list L such that for any binary sequence S, S is on the list L.
So you have to do this in steps:
Step 1: You have to choose a list.
Step 2: I choose a binary sequence.
Query: Is the sequence I chose on the list you choose?
Now, the kicker is that step 2 doesn't happen
until you've completely specified your list L. Once I've chosen a binary sequence, you can't go back and change your list.
Let me say this again.
You have to specify everything relevant about your list
before we start choosing binary sequences.
Let me give an example.
Suppose you give us a specification for a list.
We mention a binary sequence.
You come back with a new specification for a list.
In this example, you have
failed. The list must be completely specified before we start choosing real numbers.
And, incidentally, for your latest attempt, I choose the binary sequence:
...010101
that is, the sequence \{x_n\} where x_i = (1 + (-1)^i)/2 (i \in \mathbb{N}), or equivalently, x_i = 1 iff x_i is even.
