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I am looking at Theorem 2.12 in Rudin, and had a question specifically about the diagonal enumeration step as depicted in this question. My question is around what benefit does such an enumeration accrue? From the definition, I get that a set is countable if it can be put in correspondence with the set of positive integers.

However, I fail to see how enumerating the elements in a diagonal manner is different from just enumerating them by listing the elements of the first set, followed by the elements of the second set, ..., and using the same set of arguments as follow later in the proof? I'd like to think that the set of elements that are associated to the same diagonal is countable (and not necessarily finite), and thus, it appears to me that this is the same configuration as considering a sequence constructed by starting with the elements of the first set (which is countable), following this by the elements of the second set, and so on. I'm sure I am missing something elementary in my understanding, most likely related to the diagonal association being finite, not just countable, and would appreciate some enlightenment.

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indivisibleatom
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1 Answers1

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Enumerating all of the elements using the diagonalization technique allows you to pair every element of your collection to a unique natural number.

If you decided to first try and pair off all the elements of your first countable set with the natural numbers, you would never reach a point where you could start indexing the second set.

Rocket Man
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  • But, if I get sufficiently far into my indexing, might I not reach a similar point (from where I can't start indexing the elements of the next diagonal?) -- this falls back to every diagonal being countable as opposed to finite (in which case I would agree completely with the above statement). – indivisibleatom Nov 26 '15 at 02:14
  • Every diagonal is finite. Using the diagonalization method, if you give me any element of your collection, I can give you the exact number it corresponds to. Maybe easier is to pair each element with an ordered pair of natural numbers and treat the diagonalization technique as putting your elements into a grid. – Rocket Man Nov 26 '15 at 02:16