1. The density of primes in natrual numbers tends to zero if using the prime number theorem, yet a theorem states that any infinite subset of the natural numbers has cardinality of aleph_0, in which case the density is aleph_0 / aleph_0. Isn't this ratio equal to 1?  2.

3. Originally Posted by talanum1
The density of primes in natrual numbers tends to zero if using the prime number theorem, yet a theorem states that any infinite subset of the natural numbers has cardinality of aleph_0, in which case the density is aleph_0 / aleph_0. Isn't this ratio equal to 1?
no  4. Then one of the aleph_0's must act like a constant, in which case it is both a constant and a number that can tend to get bigger.  5. Why do you think things like the alephs behave like ordinary numbers?  6. Originally Posted by talanum1
Then one of the aleph_0's must act like a constant, in which case it is both a constant and a number that can tend to get bigger.
ridiculous  7. Let's not name call.

Put it this way, all cardinal numbers measure are whether two sets can be put into 1-1 correspondence with each other. This is a concept which is quite far removed from the concept of density.

Keep in mind that phrases such as "density of primes" and "cardinality of integers" have extremely precise meanings in mathematics. One cannot infer mathematical conclusions about these terms based on the everyday meaning of words like "density", etc.  8. Rigorously ridiculous? You didn't show a contradiction. Both of the concepts are properties of infinite sets.

Neither did you prove (demonstrate) they are logically incompatible.  9. Originally Posted by talanum1
Rigorously ridiculous? You didn't show a contradiction. Both of the concepts are properties of infinite sets.

Neither did you prove (demonstrate) they are logically incompatible.
Nonsense  