This is probably not an original thought, almost certainly it isn't. But it's a nice idea. List prime numbers (numbers that are divisible only by 1 and themselves):
2, 3, 5, 7, 11, 13, 17, 19, 23
Now index them as
1, 2, 3, 4, 5, 6, 7, 8, 9
Now only chose primes with a prime index, so you get "prime primes':
3, 5, 11, 17,
Now index those
1, 2, 3, 4,
Repeat and you get
5, 11
If you do this for all the infinitude of prime numbers, can you repeat forever? So that no matter how often you index them, you still get an infinite list? My answer is, yes, no matter how often you filter by indexing and choosing only prime indices, you go on forever! It kind of follows from Cantor's proof of the infinity of the natural numbers being the same, no matter how often you divide them by 2 etc. As you have a never ending supply of prime numbers, you can never whittle them down by using another infinite list, which the list of prime indices is, - infinite. The set of the subset of primes is always the same size as the set of the whole.
Infinity. Hurrah!
It would be cool to write a little computer program to do this. But obviously working with a finite subset of primes ;-)
Update - it occurs to me that you can have a hard core version of this by using the prime primes to pick out the next generation of primes.
Update 2 - I was going to call my above sequence of primes super-primes, and googled that term, to discover, that voila! Super primes are actually what I independently "discovered" here. Hurrah!
