Quote:
Originally Posted by Aaron W.
You probably need to contemplate what one is actually saying when they "raise 2 to a cardinal number power." What you're calculating is the size of the power set.
It's best understood by analogy:
The size of the power set of {1,2,3} is 2^|{1,2,3}| because for each element in the set, you have to pick from two options to create a subset. Either the element is in the set or it's not.
So the fact that you might try to combine primes in some way to create other numbers literally has no meaning. You're thinking about the number of subsets of the original set that you can create, not some other arithmetic property about the elements in the set.
I think DS is perfectly aware that 2^N is the size of the power set. His reasoning is this:
- the size of the set of prime numbers is aleph null;
- so the size of the power set of the prime numbers should be aleph 1;
- but if I can map each element of the power set to a distinct natural number that would make the PS aleph null;
- and I can make this map, since I can map each element of the PS to the natural number obtained as the product of the numbers in the set!
What's wrong in the reasoning above, as I expressed in my earlier comment, is that aleph 1 elements of the PS are themselves of infinite size. Take for instance the set of all primes except 2. This set does not map to any natural number, according to the map proposed by DS.