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"Bigger" is poor wording when discussing cardinality. I can see other comments here are struggling with this. Cardinality should not be thought of as "size." C

"Bigger" is poor wording when discussing cardinality. I can see other comments here are struggling with this. Cardinality should not be thought of as "size." Cardinality is purely a functional property. If you can define a bijective function between two sets, then they have the same cardinality. If you can't, then you take the set which has all of its elements paired, place it before the set that has unpaired elements, and there you go, you've defined a partial order over all possible sets, including infinite sets. But it doesn't really match what we usually mean by "size." The size of a set is more properly associated with measure. This is why we rely on measure to coherently formalize continuous probability. The sets (0, 0.1) and (0, 0.9) have exactly the same cardinality, but the latter has greater measure, which is why we can say an event space defined by the latter is more probable than one defined by the former. Both event spaces are infinite, but one is "bigger" than the other, in this case in a way that captures intuition about size much better than cardinality does.

contravariant 5y ago
If you were hoping to make size more intuitive by using measures then immeasurable sets kind of pose a problem. With Banach-Tarski as the ultimate example.
syops 5y ago
One can define an order relation using cardinality. Cardinality does capture a notion of size of sets that corresponds to intuition for finite sets and captures the idea of one infinite set being “smaller” or the same size as another. In general sets don’t have a measure on them in the sense of measure theory. Indeed, assuming the Axiom of Choice there are non measurable sets of the real numbers. But every set has a cardinality. And there is a well defined linear order of the hierarchy of cardinal numbers. I’m assuming the axiom of choice.
akira2501 5y ago
> Cardinality is purely a functional property. Is something like "granularity" an appropriate intuition? Both sets are of infinite size, but one has more points with which to take measurements against than the other. It isn't "bigger" so much as it is "finer?"
- jobigoud 5y ago
  But this breaks when you compare say, the set of integers vs the set of prime numbers or the set of powers of 2. The intuition say naturals are more granular and the others more sparse, but they are infinites of the same size...

contravariant 5y ago

If you were hoping to make size more intuitive by using measures then immeasurable sets kind of pose a problem. With Banach-Tarski as the ultimate example.

syops 5y ago

One can define an order relation using cardinality. Cardinality does capture a notion of size of sets that corresponds to intuition for finite sets and captures the idea of one infinite set being “smaller” or the same size as another. In general sets don’t have a measure on them in the sense of measure theory. Indeed, assuming the Axiom of Choice there are non measurable sets of the real numbers. But every set has a cardinality. And there is a well defined linear order of the hierarchy of cardinal numbers. I’m assuming the axiom of choice.

akira2501 5y ago

> Cardinality is purely a functional property. Is something like "granularity" an appropriate intuition? Both sets are of infinite size, but one has more points with which to take measurements against than the other. It isn't "bigger" so much as it is "finer?"

jobigoud 5y ago
But this breaks when you compare say, the set of integers vs the set of prime numbers or the set of powers of 2. The intuition say naturals are more granular and the others more sparse, but they are infinites of the same size...