I'm a math hobbyist, so forgive me if what I ask is silly.

I just learned that the cardinality of Reals is greater than the Naturals.

So, because of that, there can be no turing machines which generate all real numbers (ignore the fact that the turing machine would never halt).

In particular, there can be no algorithm which sequentially generates all real numbers, because, since turing machines operates sequentially, we could put them into correspondence with the Naturals, which would be a contradiction of the Cantor Theorem.

Is my thinking correct? Could you imagine what other implications would this have?

Thanks for the time.

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    $\begingroup$ Yes. What you've said is basically that every recursively enumerable set is countable, co an uncountable set can't be recursively enumerable. But there are also countable sets which are not recursively enumerable. $\endgroup$ – tomasz Sep 2 '12 at 14:50

You are correct. Another way to say this is that the set of computable reals is countable, so cannot be all the reals. In fact, in the measure-theoretic sense almost all reals are not computable.

  • $\begingroup$ Thanks a lot for the clarification. $\endgroup$ – Vinicius Seufitele Sep 2 '12 at 19:26

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