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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)

SheppLogan
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  • 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. –  Jun 29 '17 at 15:49
  • Literally the first question on the site... :| – Asaf Karagila Jun 29 '17 at 17:12

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"Dense" does not imply "uncountable": the rationals are dense but countable. Similarly, the Cantor set is uncountable but nowhere dense.

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.

Noah Schweber
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  • There are a couple meanings of dense. In the sense of topology, the cantor set is nowhere dense in the reals (and dense in itself). In the sense of ordered sets, the cantor set is a dense linear order. –  Jun 29 '17 at 15:43
  • @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 $2000\ldots$?) –  Jun 29 '17 at 16:42
  • But would you say that, ''intuitively'' dense means that if you take any sub-interval of the infinite set, you cannot count how many items there are in it ? like e.g. subset [0,0.1] from the Real numbers. – SheppLogan Jun 29 '17 at 16:59
  • @Hurkyl I think of the Cantor set as $2^\omega$; you're right. – Noah Schweber Jun 29 '17 at 18:10
  • @Machupicchu "''intuitively'' dense means that if you take any sub-interval of the infinite set, you cannot count how many items there are in it?" I would not - density gets in the way of counting points in order, but that's not what cardinality is about; cardinality is just about how many points there are. "Moreover, would you say that the reals are ''larger'' than the integers?" Yes, I would - we can prove that there is no injection from the reals to the integers. Note that "the reals are uncountable" is a precise statement. – Noah Schweber Jun 29 '17 at 18:13
  • So when we say "the reals are uncountable," we're not making a vague qualitative assertion - we are making a precise mathematical statement ("There is no injection from the reals to $\mathbb{N}$"), which has a formal mathematical proof. It's important to contrast this with vague intuitions about infinities, like "density means you're bigger," which either don't correspond to precise statements or are false. – Noah Schweber Jun 29 '17 at 18:15