PT1 said:
'Barely getting started' Poor choice of words. I think I do understand it the way you put it. I was referring to how far away from beginning I was. Infinity is open ended, (if I can call it that), but only at one extreme. The beginning of the portion of the number line to which I was referring is fixed and I guess I was thinking of how far from it I was while not getting any closer to end.Thanks.
'Generated' what I was trying to suggest is that one could take any number, 3 for example, and square it, cube it ... infinitely many times, creating an infinite group of perfect numbers to match only one prime. the number of powers of three greatly out number one which is the quantity of the number three. Then repeat for each single prime or other composite which has not yet been raised to a power.
The whole point I was getting at was that the powers spread out too fast creating the impression, to me , that there are fewer powers then other numbers. rather than the other way around which I had expected. Thanks to all.You have given me lots to digest.
Yes, this is a paradox of infinite sets. A simpler example is to consider the set of natural numbers and the set of even numbers:
##\mathbb{N} = \{1, 2, 3, \dots \}##
##S = \{2, 4, 6, \dots \}##
Clearly, we can put ##\mathbb{N}## into one-to-one correspondence with the set ##S##, in that sense, both ##\mathbb{N}## and ##S## have the same "number" of members and are sets of the same "size". More properly, they are sets of the same
Cardinality.
Equally, we can see that ##S## is a proper subset of ##\mathbb{N}##. In that sense ##\mathbb{N}## contains "twice as many" members as ##S##.
Your idea takes this a step further. If we take a prime ##p## and define the set:
##P_p = \{p, p^2, p^3 \dots \}##
Then we cleary have a set that can be put into one-to-one correspondence with ##\mathbb{N}##. Just associate ##p^n## with the number ##n##.
If we take the union of all these sets, we have:
##P = P_2 \bigcup P_3 \bigcup P_5 \dots ##
And, yet, ##P \subset \mathbb{N}##
What we've done is shown that ##\mathbb{N}## can contain a whole infinite sequence of sets, all the same cadinality as itself, and still have plenty left over: "most" numbers, after all, are not prime powers.
To put it another way, ##P## is the union of an infinite collection of sets, all the same cardinality as ##\mathbb{N}##, but still only has the same cardinality as ##\mathbb{N}##.
These sort of paradoxes were sorted out by a mathematician called Georg Cantor. If you want to look him up and read more about this.