The answer is independent of the standard axioms of set theory. One can of course add axioms to force it to go either way (e.g., trivially, the axiom “c = aleph_1” or the axiom “c > aleph_1” will do the trick, though not particularly compellingly), but people rarely see reason to bother. The way I see it, there’s not some external underlying truth of the matter. It is, as I said above, just like the observation that the ordered field axioms do not prove “There exists a square root of 2”, nor “There does not exist a square root of 2”. There are many ordered fields one can consider, some with and some without square roots of 2, and one can even consider just the ordered field axioms on their own, agnostic to the settling of this question. Similarly, there are many models of set theory one can consider, some with and some without c equal to aleph_1. There’s not some Platonic truth of the matter; there’s just where our own game-defining rules take us. The relationship between c and aleph_1 happens to not be decided by the rules of our game, any more than the canon determines Sherlock Holmes’ blood type, but I see no reason for worry over that.
So, (d) then. And it really is up to us where we want to go with it.
Thanks for that.
Good thread. Needs to be re-opened, as mod surmised. Perhaps this should be cited often. Not sure how many times that is permissible before thread lock, though.
The hypothesis that 2^ℵ[SUB]k[/SUB] = ℵ[SUB]k+1[/SUB] has been shown to imply the Axiom of Choice.
That Axiom in turn contradicts game theory’s Axiom of Determinacy and leads to the Banach-Tarski paradox. I’ve wondered whether mathematicians “believe” in the Axiom of Choice, disbelieve it, … or just believe it’s unbelievably ignorant to speak of “belief” in such things. :rolleyes:
We can always make a bigger infinity by taking the power set… but is there such a thing as a “log set” that gives us a smaller infinity? If so, doesn’t this imply that there are smaller infinities than the set of the integers? And if not, why doesn’t this work?
Most mathematicians will happily take the Axiom of Choice as part of their axiomatization of set theory, hardly ever worry about it, and, I suspect, would even say they “believed” in it.
I typically work more generally, without the Axiom of Choice. But not because I believe it to be false. That seems meaningless. I would not want to speak of “belief” in such things (any more than it’s a matter of belief whether multiplication is commutative, or x + y >= x, or a path from point A to point B entails a reverse path from point B to point A, or… it’s a stipulated or derived rule in some math-games, and it isn’t in some other math-games. There’s not some external truth of the matter; there’s just the rules we choose to explore).
It doesn’t work because there’s no such thing. How would you define “log” for sets?
If X is any well-ordered infinite set, then we can define the sequence x_0, x_1, x_2, …, as the smallest, second-smallest, third-smallest, …, elements of X, respectively, and thus inject the natural numbers into X. Accordingly, aleph_0 is the smallest well-orderable infinite cardinality.
(And every set smaller than a well-orderable set is also well-orderable. Thus, there’s no infinite set smaller than the integers)
Without the Axiom of Choice, not all sets need be well-orderable, and, indeed, not all sets need be comparable in size: one could have an infinite set which was neither smaller nor larger nor the same size as the integers. These are sometimes called “Dedekind sets”, and their sizes “Dedekind cardinals”.
::Sigh::
If only I had been given this book in junior- or high school. This is the way to teach (and how I try to do it orally): here we kind of know what the Martian thinks, and think along with him, darting ahead with a little excitement in acknowledging each previous little thrill of cognition. After reading the first 15 pages, I look forward to working with it.
But no, I went to Stuyvesant High School pre-calc. Big words, rote, calculation by plug in the numbers. Why I remember Princess Sohcahtoa, but have no idea why the sytem makes sense, but boy would I get the answers right.
It’s amazing so many clever mathematicians get out of there at all.
Today, of course, the idea is to understand what you are doing, RATHER than to get the right answer.
– Tom Lehrer, New Math (1965)
BTW, Leo Bloom, I just cited (and linked) your post in Anonymous User’s thread, as a positive reference for the Calculus Made Easy book.
Leo Bloom writes:
> It’s amazing so many clever mathematicians get out of there at all.
It’s because at least you have a chance to take first-year calculus and often another year or two of mathematics courses beyond first-year calculus. Whereas those of us who went to lousy high schools don’t even get to take calculus. And that’s no matter how brilliant you are. If you were to suggest that it might be a good idea to take calculus, you would labeled a snob and a traitor. If it were legally possible, they would have you hung for wanting to take anything that advanced.
Wendell Wagner: That’s funny, and sad, and true!
I went to a really crummy high school. But we did have a few teachers who were willing to go out on their own and make that extra individual effort. I will always remember my old math teacher who took me aside and gave me some “pre-calculus” tutoring, and lent me some very good textbooks that helped ease me into calculus.
(The biology teacher, also, was willing to talk about evolution, but only informally, in after-class talks, since the local school board had forbidden it being taught formally.)
Is this like the theory that you can divide a sphere into two infinite sets of points, then assemble them into two spheres with twice the volume of the original sphere? Because that’s what convinced me that set theory has to somewhere be seriously wrong.
ETA: and if someone says “Well, it depends on how you define ‘volume’…”, I will administer powerful electric shocks to restore them to reality.
It’s not really the same thing, but they’re both parts of measure theory. If you adopt the Axiom of Choice, you can construct pathological sets of points which behave too poorly to be consistently assigned any particular measure; you could then decompose even measurable sets (like a sphere) into unions of non-measurable sets, manipulate the non-measurable sets around, and recompose them into a measurable set with different measure than you started with.
But this has to detour through those non-measurable sets. It would also be consistent to suppose that all sets are measurable, so long as you rejected the Axiom of Choice.
But neither form of set theory is “right” or “wrong”. It can only, at worst, not actually be describing some other, external thing you were hoping it would describe. It’s still perfectly correct on its own terms. (Similarly, Euclidean geometry isn’t “wrong”, even if the space we live in has relativistic curvature. Euclidean geometry just isn’t, in that case, modelled by the space we live in. It’s an abstract game not about that space.)
Here’s a much simpler, less sophisticated example of just the same sort of thing as the Banach-Tarski paradox (the phenomenon you are referring to), incidentally: take a stick and place it on the bottom side of a an equilateral triangle. Break the stick into a shitload of individual points, then move each point straight up to the top of the triangle (where you glue them back together, if you like). Ta-da; you’ve doubled the length of the stick!
Is that disturbing to you? If not, why is the Banach-Tarski paradox more disturbing? On the other hand, if so, well, A) sorry, them’s the breaks, what’re you gonna do? but also, B) What are you gonna do? Perhaps it suggests that the way you would like to look at length (and area and volume and measure more generally) is not as some quantity assigned to arbitrary collections of points, but only as a property of “cohesive figures” which, whatever they are, are not just conglomerates of discrete points. Which is a thing you can do, if you like. After all, physical volumes are never calculated for arbitrary sets of points, but only for solid region-y regions.
Yes, you can define the account of “volume” you are interested in, and different people might be interested in different accounts; go ahead, give me my shocks, but it’s the truth.
This is in fact the Banach-Tarski Paradox septimus linked to.
The way I get my head around it is thinking of it the same way you can say there are the same number of points in a one-inch line as in the diameter of the universe. It’s a statement about infinite numbers with each point having zero dimensions. Trying to equate this to the volume of three-dimensional particles misses the, um, point. Mathematical infinities can’t be represented in physical terms: that’s the problem erik and Valmont kept having.
Any discussion of what “infinity” means necessarily involves a discussion of what “a shitload” means. 
(That said, I like this visualization! It puts the points on one line-segment into one-to-one correspondence with the points on a longer line-segment. It’s grotesquely counter-intuitive…but, as you say, what’re you going to do about it? I usually fall back to sulking and nausea…but I did get decent grades in my math classes!)
I’m still not entirely clear on what happens. If I understand correctly is it essentially saying that:
If you have a point at (2,1) and a point at (2,2) and you translate the point at (2,2) down one point in the y-direction, it’s indistinguishable (heh) from having one point. Same with rotation and many other transformations (including ones from higher dimensional spaces into lower ones).
So any single point at (x,y) can be represented as a set of all possible transformations and points (T * (a,b)) that produce (x,y). Since there’s an infinite amount of Transformation/point pairs, you can choose to transform SOME of those points in that set to another spot, thus cloning the sphere (or in fact, any shape you desire, or even “creating” objects with greater volume than sphere 1).
Is it something like that?
ETA: Okay, I realize now that I have ordered pairs, not ordered triples, so that’s for cloning two-dimensional objects, let’s just pretend everything was in three dimensions.
Thank you. That’s in fact where I thought I had been posting it, particularly since the thread was getting good especially about textbooks, and about high school. I asked the mods to repost it there to fix my error, but I guess it fell through the cracks.
As discussed in a nearby thread (the one about the rationality of square roots of integers), Indistinguishable has established a reputation for “plain-spoken” (i.e., intelligible) explanations of things mathematical. In this nomenclature, “shitload” is just another synonym for “infinite” or “infinity” 
ETA:
I think “shitload” must have something to do with this concept too.
Resurrected for this reason:
According to JRagon, in “What’s the longest thread on the Dope,” this thread is the longest thread in all of GQ.
Woo-hoo! And I’ve just augmented the thread’s personal best!