My professor gave us these properties very fast in our class and I can't find a proper explanation for them, can someone help me please?

(1) - The cardinal of the set of naturals is the same of the integers and the same of the rationals and these are enumerable sets (how is this possible? Don't the rationals have all the integers (and also other elements) and the integers have all naturals (and also other elements)??)

(2) - The cardinal of irrationals is bigger than the one for naturals and this is a non-enumerable set (this one kinda makes sense)

(3) - The cardinal of irrationals is equal to the cardinal of a set [a,b] (wait, how? Doesn't [a,b] have may more elements than the irrationals?)

Anyway, maybe the problem is that I'm not understanding the key concepts of cardinal and enumerable set... Can someone explain me please?

Stefan Mesken
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Granger Obliviate
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  • The professor probably said non*de*numerable. It's true the integers are contained in the rationals, so you could say there are at least as many rationals as integers, but you can also arrange the rationals in a sequence (not in the usual order) and this gives a correspondence with the natural numbers. So there are the same number of rationals as natural numbers. – Matt Samuel Feb 28 '16 at 00:39
  • An amusing presentation of the fact that an infinite set can have the same cardinality as certain of its *proper subsets* is [Hilbert's Hotel](https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel). – hardmath Feb 28 '16 at 15:52
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    (2) has alreay been asked [here](http://math.stackexchange.com/questions/72130/cardinality-of-the-irrationals) and [here](http://math.stackexchange.com/questions/732/proof-that-the-irrational-numbers-are-uncountable?lq=1) (3) has already been asked [here](http://math.stackexchange.com/questions/265451/the-cardinality-of-mathbbr-mathbb-q?lq=1). –  Feb 28 '16 at 17:24

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