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my son came home from school and said that there was a 1:1 correspondence between even numbers and whole numbers. Is that correct? It seems to me that even though they are both infinite, there will have to be twice as many whole numbers as even numbers. I was thinking that for an infinite set of triangles there are an infinite number of side, but the ratio of sides to triangles is still 3.

Joseph Hirsch
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    Congratulations! You have just discovered Galileo's paradox. – MJD Jan 05 '22 at 05:00
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    It's true that there is a 1:1 correspondence, namely $n\leftrightarrow 2n$. In this sense, an infinite set can have the same size as a proper subset of itself. This is one of the many properties infinite sets have that finite sets don't. – Karl Jan 05 '22 at 05:01
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    You might also be interested in Hilbert's hotel and Cantor's diagonal argument. Welcome to the study of infinite sets! – Karl Jan 05 '22 at 05:05
  • That's one of the conditions for a set to be infinite - it can be mapped 1-1 with a subset of itself. – marty cohen Jan 05 '22 at 05:10
  • Thanks. We can still talk about the ratio of whole to even numbers in any finite stretch, so I find that interesting. – Joseph Hirsch Jan 05 '22 at 05:17
  • @JosephHirsch note I just added a new answer to the linked question (as it seemed to be a duplicate), but my new answer gives two different perspectives one where your son is correct (that is the viewpoint of a one-to-one correspondence), and then I give a view point in which you are correct and that is through the lens of density. – Steven Creech Jan 05 '22 at 05:33
  • @Steven both of those are mentioned in the highly voted answer from over a decade ago, no? – Mark S. Jan 05 '22 at 13:50

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