3 ms·

Cardinalities of sets do have sensible operations like less than and greater than though - |A| <= |B| if there exists an injective function from A to B. One has

by joppy 5y ago

Cardinalities of sets do have sensible operations like less than and greater than though - |A| <= |B| if there exists an injective function from A to B. One has to check that this inequality behaves in the ways that you expect, but it is a well-defined concept.

IX-103 5y ago
It is defined in the abstract, but I think there are a lot of implicit assumptions that get lost when dealing with cardinalities which may turn out to be important. Just because a bijective/injective/surjective function exists doesn't mean that such a transform is possible when you apply it to a specific situation. If sizes were sensible then for a set A composed of "every other integer" would be smaller than the a set B composed of "2 times every integer". If we were using calculus to take the limit of the ratio of sizes of the generating functions for A and B as the source set size goes to infinity then we find that A is in fact half the size of B. If we try to simply apply cardinality to A and B we find that the sets are exactly equal, since cardinality doesn't care about source sets or limits! This can a big deal if say, you're calculating probabilities across an infinite number of possible events/event configurations. There is a big difference between a 50% chance, a 0.000001% chance and a 100% chance. I'm not saying there's no use for cardinality and infinity classes, but they can easily be misapplied to allow you to be wrong, with confidence.
- anikan_vader 5y ago
  > If sizes were sensible then… Sorry, but they simply don’t meet your (non-standard) definition of sensible. Both of the sets you mentioned can be interpreted as the set of even integers. This set is in bijection with itself (trivially), and thus is not considered to be strictly smaller than itself.

3 ms·

Cardinalities of sets do have sensible operations like less than and greater than though - |A| <= |B| if there exists an injective function from A to B. One has

by joppy 5y ago

IX-103 5y ago
It is defined in the abstract, but I think there are a lot of implicit assumptions that get lost when dealing with cardinalities which may turn out to be important. Just because a bijective/injective/surjective function exists doesn't mean that such a transform is possible when you apply it to a specific situation. If sizes were sensible then for a set A composed of "every other integer" would be smaller than the a set B composed of "2 times every integer". If we were using calculus to take the limit of the ratio of sizes of the generating functions for A and B as the source set size goes to infinity then we find that A is in fact half the size of B. If we try to simply apply cardinality to A and B we find that the sets are exactly equal, since cardinality doesn't care about source sets or limits! This can a big deal if say, you're calculating probabilities across an infinite number of possible events/event configurations. There is a big difference between a 50% chance, a 0.000001% chance and a 100% chance. I'm not saying there's no use for cardinality and infinity classes, but they can easily be misapplied to allow you to be wrong, with confidence.
- anikan_vader 5y ago
  > If sizes were sensible then… Sorry, but they simply don’t meet your (non-standard) definition of sensible. Both of the sets you mentioned can be interpreted as the set of even integers. This set is in bijection with itself (trivially), and thus is not considered to be strictly smaller than itself.