On January 27 2012 16:10 supbros wrote: It's unclear what you're trying to show. Could you tell us?
It feels like you're trying to show that the set of infinitely long binary strings is uncountable? If so, you can just use Cantor's diagonalization argument instead of the awkward procedure you used in your first post.
I wasn't trying to show anything, just laying out the confusion I had while in the shower wherein I thought I created a natural number during diagonalization which is clearly absurd.
On January 27 2012 12:57 Ninja_Bread wrote: But you're counting with integers.... that means 4 ascends 3, and 4 is equal to 0100 (not sure why you flipped the bits backwards) regardless of how many 0's you put - they don't change the value... or maybe I am missing your point?
The bits were flipped backwards because otherwise for any specific number, you'd need an infinite number of 0's to the left, which is a bit weird to write and is rather nonstandard notation.
i.e. #1 = ....00000000001 instead of #1 = 100000000.....
So shouldn't you write #1 as 1000000000000 instead of 0000000001? The only thing you're changing is the base, the same principles still apply....
I feel like you're thinking of something completely different than what you're demonstrating
On January 27 2012 09:07 EtherealDeath wrote: One might be tempted to think we just generated a natural number which is not a natural number - but wait, how can the natural numbers not be one-to-one with the natural numbers, that is, how can a countable set be uncountable?
So would you please try to explain yourself clearer....?
From what I can gather, it's flipping one of the digits value, which just offsets which number is the resultant. It's still 1:1 if I'm understanding it right though....
010 is 2, but if you flip the right 0 you end up with 011 (three) If you had 011 and flip the right digit, it becomes 010. Essentially, odd/even numbers would flip values, but it's 1:1
if you flip a digit of higher value, the difference between the paired numbers increases (001 [1] <-> 011 [3]) But it's still 1:1
On January 27 2012 15:14 infinitestory wrote: i think what you have stumbled upon is something called a p-adic number (in this case, a 2-adic) i don't know a whole lot about them, but the number you generate ends up being 001011111111.... which is equal to -12 or something like that.
Why do you ALWAYS beat me to the punch in all the math blogs . Basically that though, natural numbers will have 'tails' of 0s, whereas you're generating one with 1s.
On January 27 2012 12:57 Ninja_Bread wrote: But you're counting with integers.... that means 4 ascends 3, and 4 is equal to 0100 (not sure why you flipped the bits backwards) regardless of how many 0's you put - they don't change the value... or maybe I am missing your point?
The bits were flipped backwards because otherwise for any specific number, you'd need an infinite number of 0's to the left, which is a bit weird to write and is rather nonstandard notation.
i.e. #1 = ....00000000001 instead of #1 = 100000000.....
So shouldn't you write #1 as 1000000000000 instead of 0000000001? The only thing you're changing is the base, the same principles still apply....
I feel like you're thinking of something completely different than what you're demonstrating
On January 27 2012 09:07 EtherealDeath wrote: One might be tempted to think we just generated a natural number which is not a natural number - but wait, how can the natural numbers not be one-to-one with the natural numbers, that is, how can a countable set be uncountable?
So would you please try to explain yourself clearer....?
That is how I wrote it. Was just responding to a guy asking why I didn't write it the other way, and who clearly didn't get that it was an countably infinite sequence of sequences, and that the bit flipping is to show there is no bijection.
As for the second part, I was just saying how initially I somehow confused myself into thinking I had a new natural number, since previously I was using a natural number interpretation of the sequences. That leads to various contradictions, which is what the rest of that sentence is. But of course, the sequence that consists of the bit flips is not actually a natural number so no problem there.
EPT
. Basically that though, natural numbers will have 'tails' of 0s, whereas you're generating one with 1s.
