Thread: Countable infinte or uncountable infinite

1. Countable infinte or uncountable infinite

Hi,

I am taking a course in computability and logic at my university and am reading about countable and uncountable infinites.

According to my book the set (N X N X N X N...) - where N is a natural number - any finite number of times is countable. I can see how you can reach this conclusion, so that is not my problem.

However what happens if the set (N X N X N ... ) have infinitely many dimensions? Then it becomes uncountable I would say, as you can make a diagonalization argument, but just want to make sure, I have understood it properly.

T

2. Hello,

i) A countable union of contable sets, is countable.

ii) A finite product of countable sets is again, countable.

But, a countable product doesn't need to be countable. If X = {1, 2} then the sets of functions from N (natural numbers) to X is not countable, because, as you said, one can use a diagonalization argument. (Which means that the countable product of X is not countable)

See the first chapter of "Topology" of Munkres.

3. note on thosylve's question and maurcd's answer

Maurcd is perfectly correct. But to see this more intuitively, notice that the example of an countable product of N's:
N X N X N.......
is just N^N which has the same cardinality of 2^N which has the same cardinality of the power set of N which has the same cardinality of the real numbers which is, of course, uncountable. So if you use a diagonal argument, you are re-inventing the wheel.
Similarly, if you have a larger cardinality of dimensions, say M, then you will end up with the cardinality of the power set of a set with cardinality M.

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prove that n x n x n is countable

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