The integers and the rational numbers are listable by that criterion; I can specify “listing rules” that will eventually produce every given one of them. For the reals, that’s impossible.

Maybe I have missed something, but it appears to me that the only difference between attempting to list all of the real numbers and all of the natural numbers is that, with the former – if I am methodical about it – you will still be able to find numbers I have not listed between those I have.

It seems that it is still equal impossible to list all of the natural numbers however, owing to the fact that the list would be infinitely long. However many natural numbers you care to list I can still find one which you haven’t listed in much the same way as you can find real numbers which I have failed to list.

The only difference is that – again, you are methodical about your listing – the numbers I point out are not on your list are beyond the limit of the list rather than between numbers which are already included in the list.

i find it interesting that you gave these lists direction as if they were vectors.

theres another reason why numbers dont exist somewhere in this discussion. i can smell it.

The bottom line of that blog post will be something like this:

When we set out to axiomatize the natural numbers, we write down some plausible axioms (e.g. the Peano axioms) and hope they will characterize the natural numbers. And we always fail, because there are always structures *other* than the natural numbers that also satisfy our axioms. Nevertheless, out of all those structures (which are called “models of arithmetic”) there is one standard model which is what we know we have in mind.

When we set out to axiomatize set theory, we write down some plausible axioms (e.g. the Zermelo Frankel axioms) and hope they will characterize the universe of sets. And we always fail, because there are always multiple structures that satisfy those axioms. We call those structures “models of set theory”. But here the analogy with arithmetic breaks down, because it’s not at all clear that we can pick out one “standard model” that corresponds to what we really had in mind all along. Instead, it seems likely that our thoughts were fuzzy enough that we never had any particular model in mind to begin with.

So when we talk about “the natural numbers” we have something very particular in mind, even though we can’t characterize it with axioms. By contrast, when we talk about “the universe of sets” we have only a very fuzzy picture in mind and are not really thinking about any one thing in particular.

You can prove that the infinite set of *subsets* of the natural numbers, is larger than the set of natural numbers. The method is basically the same as the diagonal method used to prove that the set of real numbers is larger than the naturals: You line up the natural numbers in the left-hand column, and the subsets in the right-hand column. Then you can construct a new subset by making it different from the nth subset at the nth counting number — in other words, if the number n IS in the nth subset of your list, then it’s NOT in the new subset you’re creating, and vice versa. Therefore it’s different from every subset of the list.

You can generalize this method to prove that the set of *subsets* of a set, must be larger than that set. So the infinite number of subsets of the real numbers, is larger than the real numbers. Therefore, for any set, you have a set which is larger.

Here’s the paradox: Consider now the set of *everything*. All objects, all atoms that make up those objects, all real numbers — and all *sets* of things, all subsets of those sets, and all subsets of the set of subsets, and so on. Every Thing. If it would make grammatical sense as a noun, then it’s in.

Nothing can be larger than the set of everything because it contains everything. But we just proved that for any set, the set of subsets is larger, right?

I have no idea what the resolution to this paradox is. I know that, technically, the “set of everything” is not considered a “set” — it’s too big, so it’s a “class”, not a set. But I’m not sure how that terminology resolves the paradox. Can it be put in one-to-one correspondence with its subsets, or not? On the one hand, it should be at least as big as its subsets, since it contains all of its own subsets (along with everything else). On the other hand, the diagonal method proves that you can’t put anything in one-to-one correspondence with its own subsets.

One way to convince yourself that this is plausible is this: when you choose a ball, it’s got a number on it. What is the chance that the first digit to the right of the decimal point is zero? Answer: 1/10. What is the chance that the first *two* digits are both zeros? Answer: 1/100. The first three digits? 1/1000. And so on. Now what is the chance that ALL of the digits to the right of the decimal point are zeros? (That’s what it takes to have an integer). It’s got to be smaller than 1/10, smaller than 1/100, smaller than 1/1000, smaller than 1/10,000….. and the only number that’s smaller than ALL of these is zero. So that’s the chance of chossing a red ball.