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I just came across a definition of infinite sets as follows "A set is infinite if it can be placed in one-to-one correspondence with a subset of itself." It then proceeds to say that the set of rational numbers is countable infinite.

Consider the following subset of rational numbers: $X=\{1\}$ which is a proper subset of the rational numbers. Does the one-one correspondence arise because $1$ can be expressed as countably infinite ways as fractions where the numerator and denominator are the same?

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    $\begingroup$ The definition doesn't mean "with any proper subset." Just with some subset. So no, the rational numbers cannot be put in 1-1 correspondence with $\{1\}$. $\endgroup$ – Thomas Andrews Sep 12 '16 at 15:01
  • $\begingroup$ An aside: technically, this is the definition of Dedekind infinite. Assuming the axiom of choice, this is equivalent to just being infinite, but without choice you can have infinite (that is, non-finite) sets which are not Dedekind infinite. $\endgroup$ – Noah Schweber Sep 12 '16 at 16:31
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There is no one to one correspondence between the rationals and the set $\{1\}$.

The (almost correct) definition you quote says

with a subset of itself

not

with every subset of itself

You can find a one to one correspondence between the set of rationals and the subset of integers, or even with the subset "all the rationals except $1$".

The definition is only almost correct because, as @coffemath comments, the subset must be a proper subset - not the whole thing.

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    $\begingroup$ A correcter definition would mention 1-1 correspondence with a proper subset of itself. :-) $\endgroup$ – coffeemath Sep 12 '16 at 15:03
  • $\begingroup$ I see- because then every set would always be countably infinite.. $\endgroup$ – Kwame Brown Sep 12 '16 at 15:04
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    $\begingroup$ @coffeemath Edited with your comment thanks. $\endgroup$ – Ethan Bolker Sep 12 '16 at 15:40

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