# 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:** 1
**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, 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-2/825237/1 "2018-11-29T00:35:14Z")
“The rational numbers seemingly form a counterexample to the continuum hypothesis: the integers form a proper subset of the rationals, which themselves form a proper subset of the reals, so intuitively, there are more rational numbers than integers and more real numbers than rational numbers. However, this intuitive analysis is flawed; it does not take proper account of the fact that all three sets are infinite.” Doesn’t QM demonstrate there is a lower limit to existence? So shouldn’t there be twice as many rational numbers between 0 and 2 as between 0 and 1? So zero wouldn’t be infinitely small. And you could divide by it. ---
**Author:** ![griffin1977](https://avatars.discourse-cdn.com/v4/letter/g/977dab/32.png) [@griffin1977](https://boards.straightdope.com/u/griffin1977)\ **Post date:** [November 29, 2018, 12:53am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/2 "2018-11-29T00:53:38Z")
> [@EastUmpqua](#): > > Doesn’t QM demonstrate there is a lower limit to existence? So shouldn’t there be twice as many rational numbers between 0 and 2 as between 0 and 1? So zero wouldn’t be infinitely small. And you could divide by it. That is like saying there can’t be an infinite number of integers as I would run out of bricks if tried to build an infinitely large tower. The fact there is a lower limit to the physics doesn’t have any bearing the maths. ---
**Author:** ![glowacks](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/glowacks/32/5548_2.png) [@glowacks](https://boards.straightdope.com/u/glowacks)\ **Post date:** [November 29, 2018, 12:59am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/3 "2018-11-29T00:59:27Z")
> [@EastUmpqua](#): > > Doesn’t QM demonstrate there is a lower limit to existence? So shouldn’t there be twice as many rational numbers between 0 and 2 as between 0 and 1? So zero wouldn’t be infinitely small. And you could divide by it. Quantum Mechanics is a set of hypotheses about reality that, along with a lot of mathematics, makes very good predictions about what will happen with very small things. That mathematics also exists divorced from reality as well. It’s just an axiomatic system in which you can prove things given a set of hypotheses. Mathematics only describes reality when we put in hypotheses that correspond with reality. It can also describe things that are completely unrealistic and that exist potentially only in the minds of mathematicians and their work. The existence of the real numbers as a set the way mathematicians describe it does not necessarily correspond to anything in reality. But they can definitely tell you what the various properties ascribed to the real numbers mean in terms of other properties they then must have. ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [November 29, 2018, 1:06am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/4 "2018-11-29T01:06:09Z")
> [@EastUmpqua](#): > > “The rational numbers seemingly form a counterexample to the continuum hypothesis: the integers form a proper subset of the rationals, which themselves form a proper subset of the reals, so intuitively, there are more rational numbers than integers and more real numbers than rational numbers. However, this intuitive analysis is flawed; it does not take proper account of the fact that all three sets are infinite.” > > Doesn’t QM demonstrate there is a lower limit to existence? So shouldn’t there be twice as many rational numbers between 0 and 2 as between 0 and 1? So zero wouldn’t be infinitely small. And you could divide by it. Math isn’t the same as physics. There is no maximum integer, even though there is a maximum number of particles in the observable universe, and likewise, there’s no smallest positive rational (or real), even if there is a smallest possible distance in the universe. And now I see that griffin1977 has ninja’d me ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [November 29, 2018, 1:23am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/5 "2018-11-29T01:23:13Z")
And even if it were relevant, which it isn’t, quantum mechanics does not in fact state that there is a “lower limit to existence”. ---
**Author:** ![Yllaria](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/yllaria/32/3452_2.png) [@Yllaria](https://boards.straightdope.com/u/Yllaria)\ **Post date:** [November 29, 2018, 1:23am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/6 "2018-11-29T01:23:45Z")
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 (IIRC) by pairing members of both sets and then pointing to the right (the number line’s left) and saying “then they both go on forever”. It’s that “they went that-a-way” that does it. Now, if you ask how some infinite sets can be larger than other infinite sets, I’m sure that question’s been discussed here before. ---
**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, 1:37am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/7 "2018-11-29T01:37:13Z")
The straightforward answer to the question posed by the thread title is: because there is a one-to-one correspondence between them. Let A = the set of rational numbers between 0 and 1, and B = the set of rational numbers between 0 and 2. Every x in A matches up with a unique number 2x in B. Or, looking at it from the other direction, every y in B matches up with y/2 in A. The OP seems to be suggesting that there are some numbers y for which y/2 is “too small to exist.” But there is no “lower limit to existence” in mathematics. The axioms on which the real number system is based guarantee that y/2 must exist _in a mathematical sense_—which is not the same as existing within the physical universe (echoes of the recent thread [Is mathematics invented or discovered](https://boards.straightdope.com/sdmb/showthread.php?t=865580)!). ---
**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, 1:42am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/8 "2018-11-29T01:42:30Z")
It’s much worse than the OP thinks. There are as many points in one inch as there are in the entire universe! ---
**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, 1:43am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/9 "2018-11-29T01:43:27Z")
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. Think of the planck scale as a ruler with only marks for full inchs; you can’t measure 5/16" even if it exists. ---
**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, 1:58am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/10 "2018-11-29T01:58:39Z")
> [@Chronos](#): > > And even if it were relevant, which it isn’t, quantum mechanics does not in fact state that there is a “lower limit to existence”. Why can’t zero be defined as: “In July 2017, the NIST measured the Planck constant using its Kibble balance instrument with an uncertainty of only 13 parts per billion, obtaining a value of 6.626069934(89)×10−34 J⋅s.[32] This measurement, along with others, allowed the redefinition of SI base units.[33] The two digits inside the parentheses denote the standard uncertainty in the last two digits of the value.” ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [November 29, 2018, 2:18am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/11 "2018-11-29T02:18:09Z")
Because defining zero in that way would make a royal mess of all of mathematics and physics. ---
**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, 2:21am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/12 "2018-11-29T02:21:03Z")
> [@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. > > Think of the planck scale as a ruler with only marks for full inchs; you can’t measure 5/16" even if it exists. Are we sure there isn’t a fundamental lower limit below which it’s not possible to exist? And if there is a limit, why couldn’t we define it as zero? ---
**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, 2:23am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/13 "2018-11-29T02:23:32Z")
The plank scale isn’t 0 and it isn’t a limit. There is a whole field called “Lattice quantum field theory” where several people work on problems that force us to consider distances below the Plank length. > **[36th Annual International Symposium on Lattice Field Theory](https://indico.fnal.gov/event/15949/)** > > The 36th International Symposium on Lattice Field Theory gathers a large community of theoretical particle physics researchers, experts in the study of strongly interacting systems – primarily Quantum Chromodynamics, the theory describing the strong... The common, but misinformed claim that the plank scale is where “laws of physics break” is simply an artifact of oversimplification or misunderstandings. The Planck length is just the minimum measurable length. Another way to think of it is wave function cannot cycle, what energy is there? What will you measure? 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. Don’t confuse the assumptions of \*completely theoretical \*Quantum gravity research topics with our current generally accepted and tested theories. Remember _string theory_ is a guess that can’t even make predictions to test right now. While the media likes it, you cannot take _anything_ it says as applying to the real world. > [@](#): > > [The relevance of the Planck scale to a theory of quantum gravity has become > a worryingly little examined assumption that goes unchallenged in the > majority of research in this area. However, in all scientific honesty, the significance > of Planck’s natural units in a future physical theory of spacetime is only > a plausible, yet by no means certain, assumption. The purpose of this article > is to clearly separate fact from belief in this connection.](https://arxiv.org/pdf/gr-qc/0601097.pdf) ---
**Author:** ![brossa](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/brossa/32/3040_2.png) [@brossa](https://boards.straightdope.com/u/brossa)\ **Post date:** [November 29, 2018, 3:13am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/14 "2018-11-29T03:13:35Z")
> [@Exapno\_Mapcase](#): > > It’s much worse than the OP thinks. There are as many points in one inch as there are in the entire universe! 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? ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [November 29, 2018, 3:27am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/15 "2018-11-29T03:27:15Z")
> [@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? It’s not hard to find a function that maps a line segment to an infinite line. Tangent for example maps (-pi/2,pi/2) to (-infinity,infinity) ---
**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, 3:52am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/16 "2018-11-29T03:52:19Z")
> [@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? Think of two concentric circles. Draw lines emanating out from their (mutual) centre. All the points on the smaller circle intersect the ‘same’ point on the larger circle. The ‘number’ of points on the circles are the same even though one has a larger circumference. ---
**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, 3:58am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/17 "2018-11-29T03:58:04Z")
> [@KarlGauss](#): > > Think of two concentric circles. > Draw lines emanating out from their (mutual) centre. > > All the points on the smaller circle intersect the ‘same’ point on the larger circle. > > The ‘number’ of points on the circles are the same even though one has a larger circumference. With your name, am I not surprised that you brought it back to points on a circle 😉 ---
**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, 3:59am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/18 "2018-11-29T03:59:31Z")
> [@rat\_avatar](#): > > With your name, am I not surprised that you brought it back to points on a circle 😉 I was actually thinking of a 17-gon approximation. ---
**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, 4:06am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/19 "2018-11-29T04:06:57Z")
> [@rat\_avatar](#): > > The plank scale isn’t 0 and it isn’t a limit. > > There is a whole field called “Lattice quantum field theory” where several people work on problems that force us to consider distances below the Plank length. > > [36th Annual International Symposium on Lattice Field Theory (22-28 July 2018): Overview · INDICO-FNAL (Indico)](https://indico.fnal.gov/event/15949/) That would have been fun. Michigan is a long way from here… ---
**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, 4:16am UTC](https://boards.straightdope.com/t/how-can-the-set-of-rational-numbers-between-0-and-1-equal-the-set-between-0-and-2/825237/20 "2018-11-29T04:16:05Z")
> [@KarlGauss](#): > > Think of two concentric circles. > Draw lines emanating out from their (mutual) centre. > > All the points on the smaller circle intersect the ‘same’ point on the larger circle. > > The ‘number’ of points on the circles are the same even though one has a larger circumference. Why wouldn’t there be some ‘left over’ points on the larger circle? [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=2)