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The following match-up makes it clear that the set of even integers and the set of positive integers have the same cardinality(size) since it establishes a one-to-one correspondence between them: enter image description here

but I could also show that the cardinality(size) of the set of even integers is just half of the cardinality of the natural numbers by establishing the correspondence like this (each even positive integer in the second set corresponds with itself in the first set): enter image description here

so is there something wrong here?

Community
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iMath
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3 Answers3

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Sets are defined to have equal cardinality if there exists a bijection between them. There is no concept of "half the cardinality" in that sense. "Half the cardinality" only makes sense for sets with finite cardinality, where we can resort to arithmetics for this definition.

What you have shown is just that existance of a bijection does not rule out existence of a non-surjective injection.

Christoph
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  • If $A=B\cup C$ is a disjoint union, you have $|A|=|B|+|C|$ and if $|B|=|C|$ then $|A|=2|B|$. There's no reason why to make an exception for infinite sets. The real fact is that $2\omega=\omega$ for any transfinite cardinality $\omega$. – AdLibitum Sep 08 '15 at 19:29
  • Yes, we can make this formal definition. However, if we wish to translate this into OP's having "half the cardinality", we should remember that $A$ having $x$ times the cardinality of $B$ does not give us a well-defined $x$ for infinite sets. Hence all that we see is that the concept does not give rise to meaningful distinctions between sets. – Christoph Sep 09 '15 at 07:34
  • @AdLibitum For comparing the “magnitude" of two different sets , Cantor thinked that the "one-to-one correspondence" ensuring the equivalence for two finite sets are also true for two infinite sets, and even use it as an axiom to develop the set theory , right ? – iMath Sep 23 '16 at 10:02
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$\frac12\aleph_0=\aleph_0$.

You just proved it.

AdLibitum
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No, there is nothing wrong.

Unlike the finite case, for any infinite set $A$, some set $B$ being a proper subset of $A$ does not imply the cardinality of $B$ is strictly smaller than the cardinality of $A$.

So the concept of one to one bijection as a notion of size does not behave exactly as it does in the finite case. Galileo had apparently observed this. (See Galileo's Paradox.)

William
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