Cantor's diagonalization

EvLer

I am stumpt on this problem:
Use Cantor's diagonalization method to show that the set of all infinite
strings of the letters {a,b} is not countable:

Can I have a hint?

StatusX

Do you understand the argument that shows that the set of decimal numbers between 0 and 1 is uncountable? Try running that argument in base two, and you have what you want.

HallsofIvy

Do you know Cantor's diagonal method? The application here is pretty direct. Assume that the set of all infinite sequences of {a,b} is countable. Then you can put them in a numbered list. Create a string z by "if the nth letter in the nth sequence is a, the nth letter in z is b, else the nth letter in z is a". Where is z in the list?

EvLer

I guess my problem is coming up with a numbering scheme of these strings, so that I could see a diagonal. Do i always need to disprove by diagonal? I know it is called diagonalization principle but still.....
I think I understand the principle, but sometimes the prof changes a small thing and then we all are lost.
So, yeah, if someone can lead me on numbering, that would be appreciated.

StatusX

You don't need a numbering scheme. The assumption that the set is countable is equivalent to the statement that there exists a bijection between it and the natrual numbers. So if you assume the existence of such a bijection and reach a contradiction, you have disproved the hypothesis that the set is countable.

EvLer

oh, ok.
thanks
that makes sense.