# .999 = 1? **URL:** \ **Category:** Factual Questions\ **Created:** [July 31, 2000, 6:28am UTC](https://boards.straightdope.com/t/999-1/27517 "2000-07-31T06:28:03Z")\ **Posts on this page:** 20\ **Page:** 53
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [October 7, 2012, 11:30am UTC](https://boards.straightdope.com/t/999-1/27517/1046 "2012-10-07T11:30:05Z")
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. ---
**Author:** ![j\_sum1](https://avatars.discourse-cdn.com/v4/letter/j/8baadc/32.png) [@j\_sum1](https://boards.straightdope.com/u/j_sum1)\ **Post date:** [October 7, 2012, 11:40am UTC](https://boards.straightdope.com/t/999-1/27517/1047 "2012-10-07T11:40:54Z")
So, (d) then. And it really is up to us where we want to go with it. Thanks for that. ---
**Author:** ![Leo\_Bloom](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/leo_bloom/32/10377_2.png) [@Leo\_Bloom](https://boards.straightdope.com/u/Leo_Bloom)\ **Post date:** [October 7, 2012, 12:34pm UTC](https://boards.straightdope.com/t/999-1/27517/1048 "2012-10-07T12:34:55Z")
> [@Derleth](#): > > [a](http://boards.straightdope.com/sdmb/showthread.php?t=158752) > > aaaay! Good thread. Needs to be re-opened, as mod surmised. Perhaps [this](http://boards.straightdope.com/sdmb/showthread.php?t=32760) should be cited often. Not sure how many times that is permissible before thread lock, though. ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [October 8, 2012, 8:00am UTC](https://boards.straightdope.com/t/999-1/27517/1049 "2012-10-08T08:00:20Z")
The hypothesis that 2^ℵ[SUB]k[/SUB] = ℵ[SUB]k+1[/SUB] has been shown [to imply the Axiom of Choice](http://mathdl.maa.org/images/upload_library/22/Ford/Gillman544-553.pdf). That Axiom in turn contradicts [game theory’s Axiom of Determinacy](http://en.wikipedia.org/wiki/Axiom_of_determinacy) and leads to the [Banach-Tarski paradox](http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_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: ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [October 8, 2012, 10:15am UTC](https://boards.straightdope.com/t/999-1/27517/1050 "2012-10-08T10:15:11Z")
> [@Indistinguishable](#): > > aleph\_1 is NOT defined as 2^(aleph\_0) [i.e., the cardinality of the powerset of the naturals; i.e., the cardinality of the reals; i.e., the cardinality of the irrationals]. That’s the definition of beth\_1. aleph\_1 is defined as the smallest (well-orderable) cardinality greater than aleph\_0. 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? ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [October 8, 2012, 4:45pm UTC](https://boards.straightdope.com/t/999-1/27517/1051 "2012-10-08T16:45:23Z")
> [@septimus](#): > > That Axiom in turn contradicts [game theory’s Axiom of Determinacy](http://en.wikipedia.org/wiki/Axiom_of_determinacy) and leads to the [Banach-Tarski paradox](http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_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: 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). ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [October 8, 2012, 4:55pm UTC](https://boards.straightdope.com/t/999-1/27517/1052 "2012-10-08T16:55:12Z")
> [@Dr.Strangelove](#): > > 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? 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”. ---
**Author:** ![Leo\_Bloom](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/leo_bloom/32/10377_2.png) [@Leo\_Bloom](https://boards.straightdope.com/u/Leo_Bloom)\ **Post date:** [October 8, 2012, 5:14pm UTC](https://boards.straightdope.com/t/999-1/27517/1053 "2012-10-08T17:14:21Z")
> [@Derleth](#): > > [I’d be remiss if I didn’t mention (eventually) that _Calculus Made Easy_ is an introductory calculus textbook which develops the whole subject in terms of infinitesimals.](http://en.wikipedia.org/wiki/Calculus_Made_Easy) It actually predates Robinson, so it doesn’t use the hyperreals, but it’s precisely the same general idea. > > [_Calculus Made Easy_ is public domain, and it’s available for free download from Project Gutenberg.](http://www.gutenberg.org/etext/33283) ::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. ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [October 8, 2012, 9:30pm UTC](https://boards.straightdope.com/t/999-1/27517/1054 "2012-10-08T21:30:05Z")
> [@Leo\_Bloom](#): > > 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. ---
**Author:** ![Wendell\_Wagner](https://avatars.discourse-cdn.com/v4/letter/w/8491ac/32.png) [@Wendell\_Wagner](https://boards.straightdope.com/u/Wendell_Wagner)\ **Post date:** [October 8, 2012, 10:12pm UTC](https://boards.straightdope.com/t/999-1/27517/1055 "2012-10-08T22:12:23Z")
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. ---
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [October 9, 2012, 1:45am UTC](https://boards.straightdope.com/t/999-1/27517/1056 "2012-10-09T01:45:44Z")
**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.) ---
**Author:** ![Lumpy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lumpy/32/446_2.png) [@Lumpy](https://boards.straightdope.com/u/Lumpy)\ **Post date:** [October 10, 2012, 12:34am UTC](https://boards.straightdope.com/t/999-1/27517/1057 "2012-10-10T00:34:17Z")
> [@Indistinguishable](#): > > The idea is that you adopt the principle “A countable disjoint union of sets has length equal to the sum of their individual lengths” (so, e.g., [0, 1] u [2, 3] has length 2) and the principle “A point has length 0” (or other principles leading to these, though these are straightforward in themselves). If you put those two principles together, the rationals, being a countable collection of points, have total length 0. And then the remainder of [0, 1] has to contribute the rest of its length; the irrationals will be assigned length 1. > > The word “length” may not seem like the right word for this; perhaps “measure” is better. Think of it as a special case of integration: the “length” or “measure” of a subset is the same as the integral of its indicator function (the function which is 1 on that subset and 0 elsewhere). > > (Now, it may not be obvious why we should consider the integral of the indicator function of the irrationals in [0, 1] to be 1, either. For example, this function is not Riemann integrable; it’s often used as an example to motivate more general notions of integration. Again, it comes down to rule-choices like the ones I mentioned above: e.g., the integral of a countable sum of functions is the sum of their individual integrals, etc.) > [@Indistinguishable](#): > > The algebraic rationals also contribute a total measure of 0. The non-algebraic irrationals in [0, 1] have total measure 1. > > Any countable set of points has total measure 0. Each particular point has measure 0, and measure is countably additive. > > [All on the conventional account of measure, of course. Feel free to devise another.] 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. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [October 10, 2012, 12:47am UTC](https://boards.straightdope.com/t/999-1/27517/1058 "2012-10-10T00:47:29Z")
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.) ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [October 10, 2012, 1:17am UTC](https://boards.straightdope.com/t/999-1/27517/1059 "2012-10-10T01:17:00Z")
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. ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [October 10, 2012, 1:17am UTC](https://boards.straightdope.com/t/999-1/27517/1060 "2012-10-10T01:17:35Z")
> [@Lumpy](#): > > 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. 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. ---
**Author:** ![Trinopus](https://avatars.discourse-cdn.com/v4/letter/t/2bfe46/32.png) [@Trinopus](https://boards.straightdope.com/u/Trinopus)\ **Post date:** [October 10, 2012, 5:12am UTC](https://boards.straightdope.com/t/999-1/27517/1061 "2012-10-10T05:12:03Z")
> [@Indistinguishable](#): > > . . . Break the stick into a shitload of individual points . . . 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!) ---
**Author:** ![Jragon](https://avatars.discourse-cdn.com/v4/letter/j/e19b73/32.png) [@Jragon](https://boards.straightdope.com/u/Jragon)\ **Post date:** [October 10, 2012, 5:26am UTC](https://boards.straightdope.com/t/999-1/27517/1062 "2012-10-10T05:26:39Z")
> [@Exapno\_Mapcase](#): > > 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. 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. ---
**Author:** ![Leo\_Bloom](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/leo_bloom/32/10377_2.png) [@Leo\_Bloom](https://boards.straightdope.com/u/Leo_Bloom)\ **Post date:** [October 10, 2012, 5:28am UTC](https://boards.straightdope.com/t/999-1/27517/1063 "2012-10-10T05:28:53Z")
> [@Senegoid](#): > > 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. 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. ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [October 10, 2012, 7:48am UTC](https://boards.straightdope.com/t/999-1/27517/1064 "2012-10-10T07:48:17Z")
> [@Trinopus](#): > > Any discussion of what “infinity” means necessarily involves a discussion of what “a shitload” means. 😉 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: > [@Trinopus](#): > > . . . 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 . . . I think “shitload” must have something to do with this concept too. 😃 ---
**Author:** ![Leo\_Bloom](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/leo_bloom/32/10377_2.png) [@Leo\_Bloom](https://boards.straightdope.com/u/Leo_Bloom)\ **Post date:** [January 18, 2013, 2:55pm UTC](https://boards.straightdope.com/t/999-1/27517/1065 "2013-01-18T14:55:58Z")
Resurrected for this reason: According to **JRagon** , in “_[What’s the longest thread on the Dope](http://boards.straightdope.com/sdmb/showthread.php?t=679398)_,” this thread is the longest thread in all of GQ. Woo-hoo! And I’ve just augmented the thread’s personal best! [Previous page](https://boards.straightdope.com/t/999-1/27517.md?page=52) [Next page](https://boards.straightdope.com/t/999-1/27517.md?page=54)