Cardinality compares two sets using one-to-one mappings. If such a mapping exists, the two sets are equal in cardinality.
In this sense, there are as many primes as there are natural numbers. Proof: arrange the primes as an infinite series of increasing numbers. Map each prime in the series to its index in the series, which is a natural number.
This definition is mathematically simple. On the other hand, the intuitive concept of “size” where the size of the real line segment [0,1] is smaller than that of [0,2] and there are fewer primes than naturals, is much more complex to define mathematically. It is handled by measure theory, but one of the intuitive problems with measure theory is that some subsets simply can’t be measured.
If I understand correctly, there really are no actual infinities in the universe, at least not inside a finite volume (and therefore not in interaction due to speed of light limits). And as far as I can make out (someone please correct me if I’m wrong), there aren’t infinitely many Everett branches arising from a quantum fork in the sense that we can’t physically measure the difference between sufficiently similar outcomes, and there are finitely many measurement results we can see. So the mathematical handling of infinities shouldn’t ever directly map to actual events in a non-intuitive sense.
In this sense, there are as many primes as there are natural numbers. Proof: arrange the primes as an infinite series of increasing numbers. Map each prime in the series to its index in the series, which is a natural number.
Yes, I get that you can do that! I get that you can do that—I just don’t know why you should do that, instead of doing it the way that seems like the sensible way to do it in my head. What recommends this arrangement over any other arrangement?
What recommends this arrangement over any other arrangement?
Nothing at all, except that it shows there exists such a correspondence—which is the only question of interest when talking about the “size” (cardinality) of sets.
(EDIT: Perhaps I should add that this question of existence is interesting because the situation can be quite different for different pairs of sets: while there exists a 1-1 correspondence between primes and natural numbers, there does not exist any such correspondence between primes and real numbers. In that case, all mappings will leave out some real numbers—or duplicate some primes, if you’re going the other way.)
All the other ways of “counting” that you’re thinking of are just as “valid” as mathematical ideas, for whatever other purposes they may be used for. Here’s an example, actually: the fact that you can think of a way of making primes correspond in a one-to-one fashion to a proper subset of the natural numbers (not including all natural numbers) succeeds in showing that the set of primes is no larger than the set of natural numbers.
Cardinality compares two sets using one-to-one mappings. If such a mapping exists, the two sets are equal in cardinality.
In this sense, there are as many primes as there are natural numbers. Proof: arrange the primes as an infinite series of increasing numbers. Map each prime in the series to its index in the series, which is a natural number.
This definition is mathematically simple. On the other hand, the intuitive concept of “size” where the size of the real line segment [0,1] is smaller than that of [0,2] and there are fewer primes than naturals, is much more complex to define mathematically. It is handled by measure theory, but one of the intuitive problems with measure theory is that some subsets simply can’t be measured.
If I understand correctly, there really are no actual infinities in the universe, at least not inside a finite volume (and therefore not in interaction due to speed of light limits). And as far as I can make out (someone please correct me if I’m wrong), there aren’t infinitely many Everett branches arising from a quantum fork in the sense that we can’t physically measure the difference between sufficiently similar outcomes, and there are finitely many measurement results we can see. So the mathematical handling of infinities shouldn’t ever directly map to actual events in a non-intuitive sense.
Yes, I get that you can do that! I get that you can do that—I just don’t know why you should do that, instead of doing it the way that seems like the sensible way to do it in my head. What recommends this arrangement over any other arrangement?
Nothing at all, except that it shows there exists such a correspondence—which is the only question of interest when talking about the “size” (cardinality) of sets.
(EDIT: Perhaps I should add that this question of existence is interesting because the situation can be quite different for different pairs of sets: while there exists a 1-1 correspondence between primes and natural numbers, there does not exist any such correspondence between primes and real numbers. In that case, all mappings will leave out some real numbers—or duplicate some primes, if you’re going the other way.)
All the other ways of “counting” that you’re thinking of are just as “valid” as mathematical ideas, for whatever other purposes they may be used for. Here’s an example, actually: the fact that you can think of a way of making primes correspond in a one-to-one fashion to a proper subset of the natural numbers (not including all natural numbers) succeeds in showing that the set of primes is no larger than the set of natural numbers.