The Last Answer, by Isaac Asimov

Murray said, “But the odd integers can be derived. If you divide every even integer in the entire infinite series by two, you will get another infinite series which will contain within it the infinite series of odd integers.”

For S={2,4,6,8...}

S/2={1,2,3,4,..}

I used to think the set of even integers is a subset of the natural numbers.Doesnt this suggest that the reverse (the set of natural numbers being a subset of even integers) is actually true?

CurtMonash2• 12 years ago

No, not a subset, for no more reason than that 17 is a member of the latter set and not of the former one.

What you're seeing is a 1:1 correspondence between an infinite set (the natural numbers) and a proper subset of same (the even natural numbers).

pedrosorio• 12 years ago

I think what you mean is you used to think that the set of non-negative even integers was "smaller" than the set of natural numbers. The reason you can do this is because the set of non-negative even integers has the same cardinality as the set of naturals (both are countably infinite): http://en.wikipedia.org/wiki/Cardinality

The same is true of the rational numbers, by the way. There is a famous proof of the fact that there are infinite sets with larger cardinality that the naturals (the reals for example): http://en.wikipedia.org/wiki/Cantor's_diagonal_argument

biesnecker• 12 years ago

I read this every time it appears on the front page, and I enjoy it every time. Asimov remains an inspiration.

kordless• 12 years ago

It is the awareness that you are immortal that would lead to the desire to end things (assuming the concept actually exists). Logically, the voice is simply a conduit to the interconnection (nexus) of all entities. Being singley aware my own consciousness is immortal and that consciousness is also part of a greater whole are equally appealing and frightening to me. That duality is slightly puzzling and interesting.

The Last Answer, by Isaac Asimov

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