# Intuitive explanation of different types of “infinity” [duplicate]

I have been taught in some computer science theoretic courses that two types of infinities exist: dense and countable, e.g. dense (uncountable) : real numbers, countable: integers.

And that therefore dense "sort of" > countable...

I would be interested if someone develops these concepts..

Moreover, could an ''intuitive'' definition be given for ''dense'' with regards to infinities ? Also, can you disscuss if it makes sense to say thant the reals are larger than the integers ? (as it is sometimes said informally)

## marked as duplicate by Mikhail Katz, Lord Shark the Unknown, Asaf Karagila♦, Antonios-Alexandros Robotis, ArnaldoJun 29 '17 at 18:14

• You'll have to be more descriptive of the notions that your teacher was talking about; that the word "infinity" (or maybe the adjective form "infinite" which is usually more appropriate) was used in the description conveys almost no information whatsoever. – Hurkyl Jun 29 '17 at 15:49
• Literally the first question on the site... :| – Asaf Karagila Jun 29 '17 at 17:12

A set is countable if and only if there is an injective map from it to $\mathbb{N}$; this has nothing to do with density.
• @Hurkyl "The cantor set is a dense linear order" No, it's not - e.g. there is no point in between $0111111...$ and $100000...$. – Noah Schweber Jun 29 '17 at 16:24
• Ah, that was bugging me when I said it but I failed to prove otherwise. (and, I assume, you mean $0222\ldots$ and $2000\ldots$?) – Hurkyl Jun 29 '17 at 16:42
• @Hurkyl I think of the Cantor set as $2^\omega$; you're right. – Noah Schweber Jun 29 '17 at 18:10