Comment by empath75
5 days ago
how large is the set of all possible subsets of the natural numbers?
edit: Just to clarify -- this is a pretty obvious question to ask about natural numbers, it's no more obviously artificially constructed than any other infinite set. It seems to be that it would be hard to justify accepting the set of natural numbers and not accepting the power set of the natural numbers.
Only countably many of those subsets can be distinguished from other ones. So "anything I could ever imagine caring about" is surely still a countable set.
I think the argument you are trying to make rests on a pretty serious fallacy generalizing "I care about some subsets of natural numbers" (and maybe " I care about subsets of natural numbers, in general") to "I care about all subsets of natural numbers, including undefinable ones".
One could argue that infinite subsets of the natural numbers are not really interesting unless one can succinctly describe which elements are contained in them. And of course there is only a countable number of such sets.
I don't agree, but I agree it's an interesting discussion to have.
When is the set of all possible subsets of natural numbers worth considering more than the set of all sets which don't contain themselves (which gets us Russell's paradox of course), once we start building infinite sets non-constructively?
The naturals to me are a clearly separate category, as I can easily write down an algorithm which will make any natural number given enough time. But then, I'm a constructionist at heart, so I would like that.
You can construct a real number by using an infinite series so it's no less constructive than a rational function on the naturals.
Non-constructive arguments are things like proof by contradiction i.e., the absence of the negative implies the existence of the positive.
Except we can only describe those infinite series for a countably infinite number of the reals, so there are all these reals expressed by infinite series we don’t have any way to describe. Why do we need those ones? (To be clear, I realise this isn’t the current standard opinion of most mathematicians, I choose to be annoying).
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Some people (not me) would consider only countably many of those subsets to be “possible”.