# How can the set of rational numbers between 0 and 1 equal the set between 0 and 2? **URL:** \ **Category:** Factual Questions\ **Created:** [November 29, 2018, 12:35am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237 "2018-11-29T00:35:14Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [November 29, 2018, 4:28am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/21 "2018-11-29T04:28:57Z")
Quantum mechanics is a rich source of mathematics, but this has nothing to do with it. Simply put, the property of a set having the same number of elements as a proper subset of itself (call this Dedekind-infinity) characterizes all infinite sets in ZFC set theory. (Note you definitely need some sort of formal set theory even to compare how many elements various sets contain.) This all dates back to around 1888. ---
**Author:** ![KarlGauss](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/karlgauss/32/3713_2.png) [@KarlGauss](https://boards.straightdope.com/u/KarlGauss)\ **Post date:** [November 29, 2018, 4:30am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/22 "2018-11-29T04:30:49Z")
> [@EastUmpqua](#): > > Why wouldn’t there be some ‘left over’ points on the larger circle? Choose any of the left over points on the larger circle and draw a line back to the circles’ center. That line intersects with a unique point on the smaller circle. So, no points can get left over on the larger circle. (I must ask, though, if this argument would apply in all non-Euclidean geometries). ---
**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:** [November 29, 2018, 4:42am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/23 "2018-11-29T04:42:27Z")
> [@brossa](#): > > Is this true, though? I can see mapping a line segment onto another line segment, or a plane onto another plane, but isn’t this mapping a line segment onto an infinite line? A different order of infinity? You’re not mapping a line segment, but individual points, as in Karl Gauss’ circle. The methods vary slightly, but the mapping can be done to any number of dimensions. (I believe even to an infinite number of dimensions, but I may be getting in over my head.) EastUmpqua, mathematical points have no dimension. They are not physical objects. Physics has to deal with the real; math can be abstract and ideal. No physical limitations can be placed on mathematical objects. That’s how there can be perfect circles, infinite lines, and all the non-physical structures of advanced math. ---
**Author:** ![rat\_avatar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/rat_avatar/32/255_2.png) [@rat\_avatar](https://boards.straightdope.com/u/rat_avatar)\ **Post date:** [November 29, 2018, 4:49am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/24 "2018-11-29T04:49:35Z")
> [@DPRK](#): > > Quantum mechanics is a rich source of mathematics, but this has nothing to do with it. Simply put, the property of a set having the same number of elements as a proper subset of itself (call this Dedekind-infinity) characterizes all infinite sets in ZFC set theory. (Note you definitely need some sort of formal set theory even to compare how many elements various sets contain.) This all dates back to around 1888. I am pretty sure under ZFC this is consider impossible to prove or disprove without the [Axiom of Choice](https://xkcd.com/804/). But this is a bit different than the traditional form of Hilbert’s problem #1 ---
**Author:** ![rat\_avatar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/rat_avatar/32/255_2.png) [@rat\_avatar](https://boards.straightdope.com/u/rat_avatar)\ **Post date:** [November 29, 2018, 4:53am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/25 "2018-11-29T04:53:30Z")
DOH!!! I forgot the C in ZFC meant the _Axiom of Choice_ was included… Please ignore. ---
**Author:** ![KarlGauss](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/karlgauss/32/3713_2.png) [@KarlGauss](https://boards.straightdope.com/u/KarlGauss)\ **Post date:** [November 29, 2018, 5:02am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/26 "2018-11-29T05:02:18Z")
> [@Exapno\_Mapcase](#): > > You’re not mapping a line segment, but individual points, as in Karl Gauss’ circle. . . Does this circle ‘argument’ hold in non-Euclidenan geometry(ies)? I ask because it seems to me that the ‘proof’ depends on the assumption that radial lines never intersect. ---
**Author:** ![EastUmpqua](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/eastumpqua/32/6025_2.png) [@EastUmpqua](https://boards.straightdope.com/u/EastUmpqua)\ **Post date:** [November 29, 2018, 5:04am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/27 "2018-11-29T05:04:45Z")
> [@KarlGauss](#): > > Choose any of the left over points on the larger circle and draw a line back to the circles’ center. That line intersects with a unique point on the smaller circle. So, no points can get left over on the larger circle. (I must ask, though, if this argument would apply in all non-Euclidean geometries). What about the other way around. Any two lines from the center passing through any two contiguous points on the smaller circle would create and angle passing through two points on the larger circle. How many points are in that arc length? ---
**Author:** ![Measure\_for\_Measure](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/measure_for_measure/32/557_2.png) [@Measure\_for\_Measure](https://boards.straightdope.com/u/Measure_for_Measure)\ **Post date:** [November 29, 2018, 5:12am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/28 "2018-11-29T05:12:09Z")
> [@rat\_avatar](#): > > There is no “lower limit to existence” in QM either. > > The Planck scale where where the Heisenberg uncertainty principle restricts us from measuring positions. This doesn’t necessarily mean that space or numbers are discretized and or finite. > [@rat\_avatar](#): > > Spacetime is continuous as far as we know right now and yes this is even true in the Quantum world. Non-perturbative gauge theory does calculations all the time in continuous spacetime. If Quantum Loop, String Theory, or some form of quantum gravity every become testable proven theory; we will see if they can prove if it is continuous or not. Q1: Are there serious physics models that are not continuous, i.e. discrete? Q2 (not directed at anyone): There _could_ be a mathematical system with eg 1/a google[1] being the smallest unit, right? It wouldn’t include most real numbers between 0 and 1, but you could toss out paradoxes associated with infinity, right? ETA [1] or 1.8 x 10^308, which is the highest double precision value, if you want a built-in application for this quixotic system. ---
**Author:** ![KarlGauss](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/karlgauss/32/3713_2.png) [@KarlGauss](https://boards.straightdope.com/u/KarlGauss)\ **Post date:** [November 29, 2018, 5:18am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/29 "2018-11-29T05:18:26Z")
> [@EastUmpqua](#): > > What about the other way around. Any two lines from the center passing through any two contiguous points on the smaller circle would create and angle passing through two points on the larger circle. How many points are in that arc length? The arc lengths are the same; their number of points is an equal infinity (unlike that of some infinities which are ‘larger’ or ‘smaller’ than others). ---
**Author:** ![KarlGauss](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/karlgauss/32/3713_2.png) [@KarlGauss](https://boards.straightdope.com/u/KarlGauss)\ **Post date:** [November 29, 2018, 5:24am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/30 "2018-11-29T05:24:08Z")
**EastUmpqua** - How about this: take any infinite line segment. place a zero on some point on it. then place a 2 where you think it should go. Oops, you really meant to place a 1 there. Rub out the 2 and re-label the point as 1 ---
**Author:** ![rat\_avatar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/rat_avatar/32/255_2.png) [@rat\_avatar](https://boards.straightdope.com/u/rat_avatar)\ **Post date:** [November 29, 2018, 5:56am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/31 "2018-11-29T05:56:33Z")
> [@Measure\_for\_Measure](#): > > Q1: Are there serious physics models that are not continuous, i.e. discrete? As we currently lack a believable theory for quantum gravity, no. Superstring theory and loop quantum gravity may need either minimal length or discretization of space-time but it depends on who you talk to on the nature and obviously neither of those theoretical constructs are merely thought experiments at this stage. Quantum chromodynamics, being based on SR works with curved time so it is the gravity question that is driving this. But to be honest even if the gravitation existed and the standard model was correct designing an experiment to test it is a problem just due to how weak gravity is. Superstring theory typically adds strings that have an average diameter of the Planck length but this still results in quantized time (and thus space). People just don’t like numerical solutions like Lattice QCD, but they will gratefully consume the products of it. Dr. Frank Wilczek would have had his ideas end in a m Landau’s catastrophe and not a Nobel prize without it though. This is a pretty iconic animation from his prize lecture and is a product of that method. [http://www.physics.adelaide.edu.au/theory/staff/leinweber/VisualQCD/Nobel/](http://www.physics.adelaide.edu.au/theory/staff/leinweber/VisualQCD/Nobel/) String theory is analytical and “pretty” and somehow has a good publicist seems to one of the big reasons why the perception that space and time are quantized seems so common, despite a complete lack of evidence to show it is (or isn’t). ---
**Author:** ![Nava](https://avatars.discourse-cdn.com/v4/letter/n/da6949/32.png) [@Nava](https://boards.straightdope.com/u/Nava)\ **Post date:** [November 29, 2018, 6:03am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/32 "2018-11-29T06:03:50Z")
> [@EastUmpqua](#): > > Doesn’t QM demonstrate there is a lower limit to existence? No, it doesn’t. All it hypothesizes (not “demonstrates”) is that certain items have a certain amount which happens to be supertiny but not equal to zero. Items whose amount is supertiny but not equal to zero would include, among others, the mass of a single photon or the charge of an electron. Note that the charge of a neutron is still equal to zero. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [November 29, 2018, 6:16am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/33 "2018-11-29T06:16:46Z")
> [@Measure\_for\_Measure](#): > > Q1: Are there serious physics models that are not continuous, i.e. discrete? Various [lattice models,](https://en.wikipedia.org/wiki/Lattice_model_(physics)) for example. Also examples in non-commutative geometry. > [@](#): > > Q2 (not directed at anyone): There _could_ be a mathematical system with eg 1/a google[1] being the smallest unit, right? It wouldn’t include most real numbers between 0 and 1, but you could toss out paradoxes associated with infinity, right? Real computers can and do only work with some sort of floating-point numbers, not actual real numbers. ---
**Author:** ![rat\_avatar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/rat_avatar/32/255_2.png) [@rat\_avatar](https://boards.straightdope.com/u/rat_avatar)\ **Post date:** [November 29, 2018, 6:34am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/34 "2018-11-29T06:34:38Z")
> [@DPRK](#): > > Various [lattice models,](https://en.wikipedia.org/wiki/Lattice_model_(physics)) for example. Also examples in non-commutative geometry. > > Real computers can and do only work with some sort of floating-point numbers, not actual real numbers. FYI, that page needs some serious attention. The lattice gauge link is typically exponentiated as a finite jump which creates a phase factor. As exponentiation typically ends up being SU(3) matrices which allows for gauge invariance. Plus it gets rid of the hairy ball theorem. It is still a numerical solution and can have problems but that is exactly why it lets you avoid the infinitesimals. The Lagrangian is invariant under transformations. While there are discrete _gauge groups_, but neither space nor time are treated as discrete. ---
**Author:** ![Half\_Man\_Half\_Wit](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/half_man_half_wit/32/21766_2.png) [@Half\_Man\_Half\_Wit](https://boards.straightdope.com/u/Half_Man_Half_Wit)\ **Post date:** [November 29, 2018, 6:49am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/35 "2018-11-29T06:49:27Z")
> [@Nava](#): > > No, it doesn’t. All it hypothesizes (not “demonstrates”) is that certain items have a certain amount which happens to be supertiny but not equal to zero. Items whose amount is supertiny but not equal to zero would include, among others, the mass of a single photon or the charge of an electron. Note that the charge of a neutron is still equal to zero. While experimentally we only have lower limits, the mass of a photon is generally considered to be zero even in quantum theory. Quantum mechanics doesn’t generically say that there is a lower limit to every quantity. Indeed, even position, the example usually used, is continuous for a free particle (although that does get you in a bit of mathematical trouble). The Planck length’s physical significance is purely conjectural at this point. And the charge of an electron isn’t the tiniest around; quarks can have charges of a third of that (although good luck catching a quark on its own). ---
**Author:** ![psychonaut](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/psychonaut/32/4655_2.png) [@psychonaut](https://boards.straightdope.com/u/psychonaut)\ **Post date:** [November 29, 2018, 8:07am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/36 "2018-11-29T08:07:29Z")
> [@EastUmpqua](#): > > How can the set of rational numbers between 0 and 1 equal the set between 0 and 2? > [@Yllaria](#): > > I watched a math documentary, once, that explained that the infinite set of all whole numbers was equal to the infinite set of all fractions I think it’s important to pin down, terminologically, what we’re talking about here. It is **not** true that the set of rational numbers between 0 and 1 is **equal** to the set between 0 and 2, nor that the set of whole numbers is **equal** to the set of all fractions. These statements are trivial to disprove by counterexample: the number 3/2 is in the set of rational numbers between 0 and 2, but not in the set of rational numbers between 0 and 1. And likewise, 3/2 is in the set of fractions (i.e., rational numbers), but not in the set of whole numbers. If one set contains an element that another set lacks, then the two sets cannot possibly be equal. What we’re talking about is not the **equality** of sets (i.e., whether two sets have exactly the same elements) but the **cardinality** of sets—that is, whether the sets contain the same **number** of elements. ---
**Author:** ![Musicat](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/musicat/32/20189_2.png) [@Musicat](https://boards.straightdope.com/u/Musicat)\ **Post date:** [November 29, 2018, 8:42am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/37 "2018-11-29T08:42:40Z")
Although it was originally written in the 1950’s, George Gamow’s [One Two Three…Infinity](https://www.amazon.com/One-Two-Three-Infinity-Speculations/dp/0486256642/) has an excellent discussion of infinities. To a mathematician, not all infinities are created equal. I recommend reading this book. ---
**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:** [November 29, 2018, 10:29am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/38 "2018-11-29T10:29:53Z")
Hilbert’s Hotel is a classic way to introduce counting small infinities. Hilbert’s Hotel has an infinity of rooms numbered {1, 2, 3, 4, 5, … } and is full up — no empty rooms at all. A bus pulls up, and the driver announces. “I got all the rational numbers aboard. Can you put us up?” “No problem,” says the Night Manager; he’s been here before. He gets on the P/A system and announces “All guests, pack your bags. If you’re in room N, move to room 2N. You’ll find a pocket calculator next to Gideon’s Bible to help with the arithmetic.” He then turns to the busdriver. “All set. Tell your number _a/b_ to go to room number 3[sup]a[/sup]5[sup]b[/sup]; it won’t matter whether he’s in lowest terms or not. Did you bring the negative rationals? Tell _-a/b_ to go to room number 3[sup]a[/sup]5[sup]b[/sup]7.” The Night Manager goes back to his porn show, happy that he’s now got an infinite number of empty rooms in case another batch of guests arrive. ---
**Author:** ![MikeS](https://avatars.discourse-cdn.com/v4/letter/m/919ad9/32.png) [@MikeS](https://boards.straightdope.com/u/MikeS)\ **Post date:** [November 29, 2018, 12:50pm UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/39 "2018-11-29T12:50:47Z")
> [@Measure\_for\_Measure](#): > > Q1: Are there serious physics models that are not continuous, i.e. discrete? The closest one I know of is [causal set theory](http://www.einstein-online.info/en/spotlights/causal_sets/index.html), where one tries to view spacetime as a very large set of discrete events that may or may not be causally related to each other. But that one is still pretty speculative. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [November 29, 2018, 1:00pm UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/40 "2018-11-29T13:00:47Z")
> [@Measure\_for\_Measure](#): > > Q2 (not directed at anyone): There _could_ be a mathematical system with eg 1/a google[1] being the smallest unit, right? It wouldn’t include most real numbers between 0 and 1, but you could toss out paradoxes associated with infinity, right? > > ETA > [1] or 1.8 x 10^308, which is the highest double precision value, if you want a built-in application for this quixotic system. Do you mean the smallest _nonzero_ number, or the smallest number altogether? If the latter, what’s 1–1 (or any number minus itself)? If 1–1 = 1/(1.8 x 10^308), then 1 = 1 + 1/(1.8 x 10^308), and so 1 (or any other number) isn’t quite equal to itself, which, it seems to me, would lead to all sorts of paradoxes. If the former, I don’t see how it “tosses out paradoxes associated with infinity.” [Previous page](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237.md?page=1) [Next page](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237.md?page=3)