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One of the counter-intuitive things about measuring infinite sets is that the cardinality of a subset may be equal to the cardinality of the original set, contrasting with the finite world where a subset is always smaller. For example, there are just as many odd numbers as there are integers, which is key to finding room at the Hilbert Hotel, when finite intuition says there should be half as many.

My question is: is there a different way to measure the cardinality of infinite sets such that the "intuitive" relationship $S \subsetneq T \implies |S| < |T|$ holds? Does it help if you limit yourself to countable sets?

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  • $\begingroup$ No. Sets that have the same cardinality are indistinguishable from the set-theoretic perspective, so there's no hope of achieving that without having any extra structure. $\endgroup$
    – xyzzyz
    Commented Sep 26, 2017 at 18:32
  • $\begingroup$ Didn't search hard enough before posting; this is the same question as math.stackexchange.com/questions/168258/… $\endgroup$
    – Chris N
    Commented Sep 26, 2017 at 18:32
  • $\begingroup$ Although, that question asked about notions of "size" other than cardinality, whereas you are specifically asking about cardinality, in which case the comment of @xyzzyz applies. $\endgroup$
    – Lee Mosher
    Commented Sep 26, 2017 at 18:34
  • $\begingroup$ Aha. My question was overly specific, then. Thank you! $\endgroup$
    – Chris N
    Commented Sep 26, 2017 at 18:36
  • $\begingroup$ You may also want to note that being equinumerable with a proper subset is exactly what makes "infinite" infinite. That is, it is the very property of infinite sets that distinguishes them from finite sets. Without it, there would be no concept of infinite or finite, as all sets would behave the same. $\endgroup$
    – Paul Sinclair
    Commented Sep 26, 2017 at 23:02

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A different way of measuring the size of sets was proposed by Di Nasso and collaborators and the name $\alpha$-theory. This theory behaves better with respect to the intuition that a proper subset should be "smaller" than a set. The historian Paolo Mancosu in particular felt that this constitutes a kind of a revolution in foundational thinking and wrote extensively about it.

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