# A different way to measure cardinality of infinite sets [duplicate]

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?

## marked as duplicate by Ross Millikan, Lord Shark the Unknown, Nosrati, mechanodroid, Community♦Sep 29 '17 at 20:38

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.