By sequences of fixed size I mean:
say the one with length 1: $$\{ 0,1 \}$$ then length 2: $$\{ 00,01,10,11 \}$$ so let the squence be :
$$F =\{\{0,1\}^k : k \in\mathbb{N}\}$$
Let the sequence with infinite length be:
$$I = \{ \{0,1\}^k : k = |\aleph_0| \}$$
so this is what I am confused about, its very strange to me that $F$ does not contain every bit sequence that $I$ contains because, if we let k go to infinite, then F should contain everything $I$ has. But at the same time we restricted k to be finite, however, as k increases unboundedly it should eventually have all the elements of $I$. I know that is not true for some reason. However, its just counter intuitive for me, what is a precise or rigorous way to make the distinction between these two sets? Are they the same? I suspect either they are the same size or $F$ is larger, but I am unsure how to prove this.
Is the only way to make a distinction of these two is by making a precise bijection of the natural numbers to F and then showing by diagonalization that natural numbers don't bisect to $I$?
What exactly is this bijection of natural number to $F$?
The thing that was confusing me about a candidate bijection (from reals) to $F$ was that, for each finite k, we would have the sequence $0^k$. Like 0, 00, 000, 0000, 0000 etc. If I were to do the bit representation of each natural number that would map a bunch of numbers but would leave these sequences that are similar to another without a pair, right? If I decide to pair say:
- 0 to 0
- 1 to 1
- 2 to 10
- 3 to 11
- 4 to 100
- ... etc
i.e. f maps a real number to its binary number representation. Then this mapping seems to not cover all of $F$, but we already ran out of elements of the natural numbers to map to bit sequences in $F$, therefore, $F$ has to be uncountable. Is that right?
Then if that is right, then the mapping function I defined, what does it actually map to then? Does its range have a special name? Its not all the sequences of fixed length nor of infinite length... cuz thats what $F$ and $I$ are...
Is $F$ countable or not?