However, you can’t really exhibit an uncountable order, but for other reasons (not involving the axiom of choice I think): to actually have uncountably many elements of a set, you need elements that *individually* can’t be written down. But that’s not an issue with ordinals. There are only countably many finite strings, and they’re too few for ordinals *and* for reals.

Still, if we start looking at ordinals, we see (if I don’t screw up details, I read this ages ago in Bertrand Russell’s “Introduction to mathematical philosophy”, now online at http://people.umass.edu/klement/russell-imp.html):

First, a progression (sequence isomorphic to naturals):

0, 1, 2, 3, …

Then, a progression of progressions:

ω, ω + 1, ω + 2, ω + 3, …

ω·2, ω·2 + 1, ω·2 + 2, ω·2 + 3, …

ω·3, ω·3 + 1, ω·3 + 2, ω·3 + 3, …

…

Then, a progression of progressions of progressions, repeating the above from each of ω², ω³, …, ω^ω

ω·ω, ω·ω + 1, ω·ω + 2, ω·ω + 3, …

ω·ω + ω, ω·ω + ω + 1, ω·ω + ω + 2, ω·ω + ω + 3, …

ω·ω + ω·2, ω·ω + ω·2 + 1, ω·ω + ω·2 + 2, …

…

Much more…

…

What you’ll see is there are too many elements without an immediate predecessor (I’ll guess an uncountable number of them), so even if each natural-like sequence (Russell calls them *progressions*) is countable, there’s an uncountable number of progressions.

Regarding ordinals that can’t be written down (which I just learned about), see https://en.wikipedia.org/wiki/Large_countable_ordinal.

]]>I cant understand why

]]>And a more useful reply than many of the others here, because people who are familiar with order types and transfinite ordinals are likely to already know the answer to the question.

]]>Basically, the trick is to define x < y just if x is built at an earlier level L_\alpha of the constructable universe than y or they are built at the same level and (bootstraping off the wellordering of finite sequences of L_\alpha induced by Con(ZFC + There is no definable well ordering of the reals) (though I’m a little fuzzy on moving from there is no projective well ordering of the reals to there is no definable well ordering but I think you do some crazy tricks with countable models).

]]>“It’s very much like the Banach-Tarski paradox: we can say that there’s a way of doing it, only we can’t actually do it in practice. In the B-T paradox, we can say that there is a way of cutting a sphere into these strange pieces – but we can’t describe anything about the cut, other than saying that it exists.”

Are you sure you are correctly characterizing the result here? While I am quite familiar with nonconstructive existence proofs, I seem to recall that the original paper DID in fact offer a construction, as have subsequent publications. So what do you mean by “we can’t actually do it in practice”? If you mean that we cannot accomplish this with a physical object, that is totally different because the proof based on Axiom of Choice uses a decomposition of a set into nonmeasurable sets, which do not occur in chemical matter because, as a combination of atoms, chemical matter is discrete and measurable. But we can do this with continuous sets, if we assume the Axiom of Choice, which counts in mathematics as “we can actually do this in practice.”

(Fun little thought: If it turns out that spacetime itself is discrete, then what we’re left with is a procedure that is possible in an idealized world of continous mathematics, but poses no crisis in the empirical world of discrete reality, but the jury is still out on whether spacetime is discrete or continuous.)

]]>There’s no contradiction because while you could pick “least” elements from R and index it that way, you’d have to do it an uncountable number of times. This is why the Axiom of Choice (equivalent to WOP) makes people uncomfortable: it involves not only “making” infinitely many “choices” but sometimes making uncountably infinitely many choices. Computationally, this is absurdity, but AoC only claims that a suitable mathematical object exists.

Now, where it does get weird is if the Continuum Hypothesis is true and |R| = 2^|N|. If CH is used (and it’s formally independent of ZFC, which means that a valid mathematics exists with it, and another valid mathematics exists without it) then we have a bijection between R and w_1, which is the least uncountable ordinal and the (uncountable, by construction) set of all countable ordinals. Thus, we can index R where each real number has a countable index, even though there are uncountably many of them. (In the same way, we can index Z or Q where every object has a finite index, but neither set is finite. The logic is similar insofar as the set of finite numbers is the smallest infinite ordinal.)

So, here’s where gets interesting. Let (R, <) be a well-ordering. Pick *any* real r. If CH, then x r than r > x, no matter which r you picked. This contradicts our intuition (“mediocrity principle”) about “random” choices from sets, insofar as if we picked a random element x of a totally ordered finite set, we can provably expect there to be about as many a > x as b < x. And there is no such thing as a uniform random choice from a countable set, so we don't have that problem with countable sets. But we start getting weird results (that don't matter in practice, but exist set-theoretically) in probability if we start talking about, e.g. "random reals".

]]>Is that right? Seems to me it’s 2 thousand trillion kilograms. I know “trillion” used to have a different meaning, at least to the Brits, but that was bigger, not smaller. How about “2 million gigatonnes?”

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