The cardinality of the number of primes in total is countably infinite. That is they can have a bijection with the
natural numbers. Consider listing them with the natural number as an index reference and the prime as the
item. The cardinality result is then obvious.
However I am not exactly sure at this time as to whether that was the question. The icon has obscured it to some extent,
also I think more words would have been a good idea. Did you mean some property of subsets of primes?
I think I might have just have thought about what is not obvious about it - I presume you haven't done the theorem
concerning cardinality of rationals being the same as the cardinality of integers and natural numbers. In which case
yes intuitively the primes may seem like a smaller set than the natural numbers, but it is known to be not finite,
so how could we have a smaller set? (I don't think we can)
The strange aspect of the Cantor/Schroeder/Bernstein results is/are that the real numbers have higher cardinality,
that is a strange result after the natural/integer/rational result. There are more real numbers than integers to the
extent that you can never list a complete set of them, or something like that - rephrase that more precisely perhaps.
I cannot remember how you prove the result with the reals, but the rational proof draws a diagram involving a sort
of rotation around the 2 dimensional number plane in which the p/q rationals in simplest form are "listed" in a
systematic manner. The proof involving reals being higher in cardinaility involves a method which I have a vague memory
of involving using the fact that there are an infinite number of numbers available after each point in an infinitely accurate
decimal representation. I think it uses a contradiction argument in a generalized listing to prove that some items must
have been missed out and that therefore no such listing can be made.
Compared to that the prime number set cardinality is quite simple which is why I wondered whether there was a result
involving finite cardinality of subsets of the primes constrained somehow, but there is not enough information in the
question to ask such a thing.
Last edited by SteveB (2013-07-31 04:29:55)