3

Why can't you use Cantor's diagonalization argument to prove that the integers are countably infinite? i.e.

1: 12345....

2: 42345....

3: 56903...

4: 46234...

5: 23421...

etc.

Then we could create a new integer by adding 1 to each number in the diagonal, so the new integer would be 23042. Where did I go wrong in my reasoning?

Marc van Leeuwen
  • 119,547
birna
  • 35
  • How do you know that $23042$ doesn't show up later? I mean, you would have to show that the number you construct is not in the complete list. –  Feb 04 '16 at 17:09
  • If you try this for the standard enumeration of the natural numbers $0,1,2,3,4,\ldots$ (trying to show it cannot exist), then you will rapidly notice that most numbers on the list do not even reach the diagonal. This is not the essential problem (which is that your "extra natural number" is not one), but you should have tried this obvious example and notice the difficulty. A similar argument does show that there are uncountably many $10$-adic integers. – Marc van Leeuwen Feb 05 '16 at 14:49

1 Answers1

7

You probably meant to ask why the argument can't be used to prove that the integers are uncountably infinite (since they are in fact countably infinite).

The problem with your argument is that you're not actually constructing an integer, since you're adding $1$ in each digit, so the thing you're constructing has infinitely many digits. If you stop at some point, you can't exclude that the integer you obtained occurs later in the list.

joriki
  • 242,601
  • What if instead of adding 1 to each digit along the diagonal, I just chose some other digit randomly (that wasn't the current digit in the diagonal)? – birna Feb 04 '16 at 17:13
  • Also, why is it not certain that the number created along the diagonal might occur later in the list for integers, but not for reals? – birna Feb 04 '16 at 17:15
  • @birna: It doesn't matter how you change the digit, by adding or randomly -- as long as you change each digit, you either have to stop or construct an infinite string of digits. The situation is different for reals because an infinite string of digits defines a real number, but it doesn't define an integer. So you don't have to stop for reals. – joriki Feb 04 '16 at 17:18
  • 3
    So essentially, you cannot change an infinite amount of digits along the diagonal, because integers cannot have an infinite number of digits (as that would approach infinity). Is that the correct understanding? – birna Feb 04 '16 at 17:22
  • 1
    @birna: Yes, it is. – joriki Feb 04 '16 at 17:28