- #1
Seasons
Firstly, thanks to everyone who participated in my last thread. It helped a lot! This will be the only other topic I can think of posting in physics forums, because, honestly, I don't know very much.
I remember sitting down one time and thinking I was quite brilliant when I started to make a binary tree; one trunk starting with all the expansions starting with 1, and another for all the expansions starting with zero. I was convinced I had found them all! Then someone pointed out to me that while every number was being slowly constructed, it kept changing places on the list, therefor, I wasn't actually, listing them, in fact, they are never listed. When I saw this, I realized, "oh, I guess I'm not so brilliant."
One thing that struck me however was the diagonalization argument, and constructions upon the diagonalizations (such as adding 1 to each digit). It's certainly not a proof that all the reals are being counted, but it is a proof that diagonalization isn't a disproof. Why? Because you can just make a new list of all the diagonalizations for each regular list, and then the diagonalizations of those diagonalization lists. This disproves the diagonalization argument, but doesn't in itself prove that all the reals are being counted.
From this, a new argument must be constructed to better understand what actually is a disproof that the reals cannot be listed.
Lets say you have lists:
List 1
Diagonalizations of List 1
Diagonalizations of diagonalizations of List 1
etc…
List 2
Diagonalizations of list 2
Diagonalizations of diagonalizations of List 2
etc...
I remember sitting down one time and thinking I was quite brilliant when I started to make a binary tree; one trunk starting with all the expansions starting with 1, and another for all the expansions starting with zero. I was convinced I had found them all! Then someone pointed out to me that while every number was being slowly constructed, it kept changing places on the list, therefor, I wasn't actually, listing them, in fact, they are never listed. When I saw this, I realized, "oh, I guess I'm not so brilliant."
One thing that struck me however was the diagonalization argument, and constructions upon the diagonalizations (such as adding 1 to each digit). It's certainly not a proof that all the reals are being counted, but it is a proof that diagonalization isn't a disproof. Why? Because you can just make a new list of all the diagonalizations for each regular list, and then the diagonalizations of those diagonalization lists. This disproves the diagonalization argument, but doesn't in itself prove that all the reals are being counted.
From this, a new argument must be constructed to better understand what actually is a disproof that the reals cannot be listed.
Lets say you have lists:
List 1
Diagonalizations of List 1
Diagonalizations of diagonalizations of List 1
etc…
List 2
Diagonalizations of list 2
Diagonalizations of diagonalizations of List 2
etc...