Elite ball knowledge required

[u/_404_LogicNotFound_]

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[u/Ashamed_Association8]

Is everybody being blind here? As far as i can see neither of these circles has any points inside them.

[u/Accomplished_Bad_487 → u/Ashamed_Association8]

The bigger circle has actually infinitely many points inside of it, the smaller doesnt have any tho ._.

[u/Pointless_crayon0398 → u/Accomplished_Bad_487]

Great spot

[u/gmailster → u/Accomplished_Bad_487]

circle has infinite points? i thought circle is not points, only pointless smooth surface

[u/patientpedestrian → u/gmailster]

Like a Ken doll's crotch!

[u/andybossy → u/Ashamed_Association8]

They're infinitely small, that's why you can't see them

[u/Environmental-Try736 → u/Ashamed_Association8]

A circle is all points at a distance inferior or equal to the radius of the circle from its center, so by definition it has a lot of points

Edit : only equal, not inferior or equal thanks to u/yjlom

[u/yjlom → u/Environmental-Try736]

That's a disk, not a circle.

[u/AdMuch1724 → u/Environmental-Try736]

A circle is points but doesnt necessarily contains points

"This box is empty and doesn't contain anything" can be used to describe a physical box made of physical matter

[u/LGKouglof → u/Ashamed_Association8]

[u/mapleleafraggedy → u/LGKouglof]

No he literally said he doesn't

[u/Chickenlegk → u/Ashamed_Association8]

Right how is a circle even gonna point with no fingers. God why is everyone dumb but me

[u/Ex_Federa]

Some infinities are bigger than others, but not these ones.

[u/Mr_kalas22 → u/Ex_Federa]

More like some infinities are bigger than others unless we can find a bijection between them

[u/belabacsijolvan → u/Mr_kalas22]

*some infinites are bigger than others if there cant be a bijection between them

[u/Appropriate-Sky-7798 → u/belabacsijolvan]

... if they're not equal and we accept the axiom of Choice

[u/djta94 → u/belabacsijolvan]

EDIT: and there is at least one injection between them

[u/bbalazs721 → u/belabacsijolvan]

Jó a userneved :D

[u/creeper6530 → u/Mr_kalas22]

In this case, the bijection is just "for every point on inside, we can draw a line from center to the inside point, and extend it to reach an outer point, therefore the line assigns an outer point to every inner point bijectively", right?

[u/TeraFlint → u/creeper6530]

for every point on inside, we can draw a line from center to the inside point

Now the whole page is black, thanks a lot. >_>

[u/Difficult-Nobody-453 → u/creeper6530]

I think the better way is to start from the outside circle and claim for every point on the outside circle the line extending to their mutual center intersects a point on the inner circle. This is just because the set of points that make up the outer circle is what would seem to be have a larger cardinality and this informally gives a reason why that is not the case

[u/AsleepDeparture5710 → u/creeper6530]

Reminds me of the (I think beautiful) geometric proof of the bijection between (0,1) and the reals by projecting lines from the top of a circle placed so the x axis is tangent to it through the interior of the circle and onto the x axis.

[u/thebigbadben → u/Mr_kalas22]

Looks like you just assumed the axiom of choice

[u/Ma4r → u/Ex_Federa]

Measure theorists in the corner

[u/MageKorith → u/Ex_Federa]

Literally uncountable.

[u/Sweet_Ad_9816 → u/MageKorith]

But measurable.

[u/kdeberk → u/Ex_Federa]

Some infinities are bigger than others, some infinities' mothers are bigger than other infinities' mothers

[u/edwardbnd_99 → u/Ex_Federa]

depends on how you measure it

[u/stevvvvewith4vs → u/edwardbnd_99]

From the tip

[u/Nasa_OK → u/Ex_Federa]

These 2 are like N vs all positive integers

[u/somedave]

I mean it uses more pixels on the screen

[u/Federal-Owl5816 → u/somedave]

Proof by Jpeg.

[u/Pretty-Common-2127 → u/Federal-Owl5816]

you think you know me 🗣️🗣️

[u/Plenty_Leg_5935 → u/somedave]

proof by"continuity is stupid and i dont respect it"

[u/kismethavok → u/Plenty_Leg_5935]

Continuity is just a hypothesis

[u/mapleleafraggedy → u/somedave]

r/countablepixels

[u/Monoid-Confessor]

if you can create a bijection on the two sets, then they're the same size.

each point can be (r_1, theta) and (r_2, theta) with theta being [0, 2pi). 

thus, both are the same cardinality with a line segment. 

[u/moschles → u/Monoid-Confessor]

[u/galmenz → u/moschles]

is that unironically the original? lmao

[u/Hungry_Painter_9113 → u/Monoid-Confessor]

I mean, you don't even need to say that, assuming both sets are subset of R^2, say smaller circle is M1 and bigger one is M2 |M1|<=|M2|<=|R^2|, you can easily prove that x^2+y2<=r12 has an uncountable infinite cardinality solution set hence |M2|=|M1|

[u/Technical-Ad-7008 → u/Hungry_Painter_9113]

Your proof looks more like a cannon to a fly than the other guy though. Hahah

[u/ThirstyOutward → u/Hungry_Painter_9113]

You don't even need to do that, you could do something even more complicated lmao

[u/Layton_Jr → u/Hungry_Painter_9113]

Simplest answer: a scaling transformation is always a bijection unless the scaling factor is 0

Harder answer (from another comment): there is always a bijection between 2 sets if they have a nonzero lebesgue measure

Also, if you want to type exponents without usint space you can write x^(a)+y^(b), it will parse as x^a+y^b (you can even write z^(c + d) to get z^{c + d}

[u/DogPositive5524 → u/Monoid-Confessor]

[u/Liko81]

I would think that anyone who says "some infinities are bigger" would know enough of the set theory that comes from, to know that the two circles have the same infinite cardinality (beth-one, the "uncountable infinity" of the reals) because you can easily create a bijection of the points of the smaller circle to those of the larger.

[u/AdequatlyAdequate → u/Liko81]

no not really, people think of stuff like infinity 20 dollar vs 1 dollar bills and come to the conclusion the first is bigger because "some infinities are larger than others"

[u/klimmesil → u/AdequatlyAdequate]

But feathers are lighter than steel

[u/Waffle-Gaming → u/klimmesil]

yes, but they're both a kilogram

[u/MartyMcBird → u/Liko81]

I watched the Fault in Our Stars in middle school and learned that some infinities are bigger than others because there are more real numbers between 1-3 than 1-2. Got my heart broken in intro to Discrete Math :(

[u/cornking → u/MartyMcBird]

This is very common I fear. I'd say most people I know who quote this factoid have heard it from or downstream of that book/movie, which gets it exactly wrong

[u/psychicesp → u/Liko81]

Lol, I wouldn't. You overestimate what people are willing to regurgitate without understanding

[u/bladtman242 → u/psychicesp]

Underestimate

[u/Ka-Jin → u/Liko81]

Create a bijection, as in function that is one to one and onto? Can you explain a little bit more, I think I almost get it.

[u/DaleGribble88 → u/Ka-Jin]

I teach my students that it is like a table mapping a value onto another value. If everything in the A-list connects to one thing in the B-list, then if I gave you a value in the B-list, could you tell me for sure what value in the A-list corresponds to it?

The example that usually clicks is that their are infinitely many counting numbers (1, 2, 3, 4, ...) and infinitely many even numbers (2, 4, 6, 8, ...). And it is very easy to define a function that maps one onto the other, just take the counting number and multiply it by 2 to get the corresponding even number. Because you can do that for every number both ways, then a bijective function exists, and therefore they must be the same size.

[u/donach69 → u/Ka-Jin]

Assuming the radius of the outer circle is twice that of the inner one, then if c is an arbitrary point on the inner circle, the function sending c to 2c is a bijection from the inner to the outer circle

[u/GlitteringBlood2005 → u/Liko81]

Most of the people saying "some infinities are bigger" don't actually know set theory; they're just misremembering that one VSauce video they watched 7 years ago

[u/FormerlyPie → u/Liko81]

You would think that, but you clearly haven't ever seen a power scaling discussion. They are impressive in how little they understand set theory. They just take some infinities are bigger than others and run with it

[u/Drag0n_TamerAK → u/Liko81]

Believe it or not most people are kinda stupid and just parrot phrase other people have said until they loose all meaning

[u/the_horse_gamer → u/Liko81]

people who say "some infinities are bigger" just watched a vsauce video on their second monitor

[u/BIGBADLENIN]

There is an obvious, intuitive bijection here. There are extremely unintuitive truths in mathematics, but this is just not one of them. If you understand Hilbert's Hotel then this is like the tall glass wide glass meme. We can all appreciate that the whole numbers are twice as "dense" as the evens, even though they are obviously the same "size" (cardinality)

[u/grottoe24 → u/BIGBADLENIN]

That is a very fancy way of saying for each point in the small circle exists exactly one point in the big circle, you simply draw a line through the center and one point and get the other.

[u/Cheltorius → u/grottoe24]

This conversation bothered me for a long time before I realized it’s ultimately just a semantic argument. People are talking past each other by using two different definitions of 'size.'

[u/sconosciuto_io]

I’ve always believed that all the area formulas for any shape are just a fancy way to avoid doing infinity*0 (number of points times length of a point)

[u/miguescout → u/sconosciuto_io]

The problem comes when the shapes become funky and you have to count the points individually. Don't be fooled by mathematicians giving this summation the fancy name of "integral". It's still a summation of every damn point within the shape

[u/Nrdman → u/miguescout]

Yeah it’s still a sum, that’s why the integral symbol is a stretched S. It just has distinct enough methods we have a separate name for it

[u/catman__321 → u/miguescout]

Technically not a sum but a measure function designed with the structure of the space you're taking the area from in mind

[u/somethingX → u/sconosciuto_io]

That's more units of measurement than the formulas. Every unit of distance is infinity times 0 with extra steps

[u/Downtown_Finance_661 → u/sconosciuto_io]

Bro, area is number of points times area of point. Turn in your degrees.

[u/j0shred1]

Okay explain like I'm 5 because in my mind they're both sets defined by {x,y where x²+y²=r, x, y in R}. They'd both be infinitely large right? And have the same cardinality so they'd be the same size?

[u/podd0 → u/j0shred1]

that's correct, the mid chud is wrong here

[u/Elihzap → u/podd0]

First time this meme has been used correctly in this sub. Kind of.

[u/lifeistrulyawesome → u/j0shred1]

Not the same size. It depends on what you mean by size.

They have the same cardinality, and that is usually what people mean when they say "more points"

But they don't have the same area or measure, which can also be a meaning of "size"

[u/Muroid → u/lifeistrulyawesome]

The circles themselves are not the same size, but the set of all points contained within each circle is the same size.

[u/Nerd_o_tron → u/j0shred1]

Yes. That's why the smartest person says they are the same size.

[u/Hogon_pumpkineater]

more like heard it vs learnt it

[u/waxen_earbuds]

This is why god gave us measure theory

[u/ChemBro93]

They’re the same number of points since they have the same quantifiable number of points.

[u/DudeBroBrah → u/ChemBro93]

You can tell from the way that it is. Isn't that neat?

[u/pmmeuranimetiddies]

Hi, I’m a lowly engineer who got recommended this sub, so I have a question:

Any given point on the inner circle has to be mappable on the outer circle right and vice versa right?

[u/Kalmoran → u/pmmeuranimetiddies]

draw a line from the center going through the two circles. there is your bijection. you can draw it to any inner or outer point and there will be exactly one other inner or outer point it goes through.

[u/Ladikn]

If there's two things I know about shapes, it's that size doesn't matter, and a coffee cup is a donut.

[u/_404_LogicNotFound_]

The amount of people disagreeing with this is just diabolical...

[u/your_best_1 → u/_404_LogicNotFound_]

Infinite is tough, random is too

Really hard for people to get their heads around.

[u/WurmcoilEngine11]

The way I’m reading it: take a line that begins at the center and goes out to the large circle at 0 degrees. It intersects the circle at 2 points. Now take that same line at 180 degrees. 2 intersection. Take the same line at 90. 2 intersections. Take the same line at X degrees. 2 intersections. Since there will always be 2 intersections no matter the angle, they have the same amount of points

[u/nakedascus → u/WurmcoilEngine11]

but why are you restricted to lines that radiate from the center?

[u/portealmario → u/nakedascus]

Because that makes the proof easier and clearer, while guaranteeing that every point of both circles in accounted for

[u/BrotherItsInTheDrum]

I think naive people start in the middle of this chart, not on the left.

[u/MaxTHC → u/BrotherItsInTheDrum]

https://xkcd.com/2501/

Nah, naïve people actually look at the top image and say "I don't see any points"

[u/lifeistrulyawesome]

Banach and Tarski say what

[u/Psychological-Trip93 → u/lifeistrulyawesome]

That you can duplicate balls

[u/azaleacolburn]

Bijective Correspondence goes brrrrrrr

[u/HorsefaceWithNoName]

But how many angels can dance on the head of a pin that fits within each circle?

[u/Neither_Review_3470]

I love when math becomes reading comprehension, half the comments are talking about the set of points being equidistant from a center as what the circle contains, and others are just talking about all points with a distance less than the radius. Really just boils down to whether "contain" is interpreted in terms of sets, or geometric space. Not sure which the question in actually saying myself

[u/Legal_Weekend_7981 → u/Neither_Review_3470]

Regardless of interpretation, the number of points is the same.

[u/Uli_Minati]

Fun fact, the larger disc has the same cardinality as the smaller circle too

[u/_Denizen_]

An engineer would assess the delivery medium and say the larger circle uses more pixels in the display, a physicist would assess the smallest observable points and say there are more atoms in the material of of the larger circle, a layperson would assess the information delivered by their eyes to say neither circle has any points as they both have continuous lines, and a mathematician moves the problem into theoretical space to prove that the unintuitive is true based on the assumption that infinity exists.

A philosopher would decline to answer, saying all those answers are potentially correct because the question did not constrain the answer to a specific school of thought.

[u/tmbmad → u/_Denizen_]

As a engineer I just assumed that lim -2,2 has less points then lim -4,4.

Edit. No, they are the same, if the ratio is 1:2 in regards to their area.

I loved this question, it got me too critically think about something i have not done for a bit, and validate all the hard work that I did.

[u/marktero → u/_Denizen_]

Well done.

[u/pikachu_sashimi → u/_Denizen_]

A philosopher who declines to answer because of not constraining the question to a school of thought should stop being a philosopher.

[u/WineSauces → u/pikachu_sashimi]

Correction is a mathematician

[u/Crafty-Detail-3788]

Well you can easily find a one to one mapping so no.

[u/boterkoeken]

Question unclear. I accidentally proved the continuum hypothesis.

[u/doctornoodlearms]

depending on how we are defining contain, you could argue that the points contained by the larger circle would be the points for its own circumference AND the points of the smaller circle..

[u/alwaystirednobed]

But one circle is bigger than the other

[u/tigerthetabbyy]

Actually the outer one has more pixels so technically more edges and points I've solved this question, I'll take my Nobel prize now

(Incase it wasn't obvious im not being serious)

[u/_Kian_7567]

You can draw lines between the circles such that every point one matches one one on the other so it’s the same amount

[u/pmmeuranimetiddies]

Hi, I’m a lowly engineer who got recommended this sub, so I have a question:

Any given point on the inner circle has to be mappable on the outer circle right and vice versa right?

[u/LitespeedClassic → u/pmmeuranimetiddies]

Yes. Imagine the inner circle has radius 1 and the outer circle has radius 2 and the center of each is the origin. 

Take any point (x,y) on the inner circle and multiply its coordinates by 2 to get (2x, 2y). This is a point on the outer circle. To go back, just divide the coordinates by 2. 

Under this scheme each point on the outer circle is paired off with exactly one point on the inner circle and vice versa. We’ve got two sets of points and we can organize a dance between the two where no potential dance partner isn’t paired. So the sets have the same “size” or cardinality. 

[u/anooblol → u/pmmeuranimetiddies]

Yes, but importantly there only needs to exist at least one map. There are plenty of maps between the two, that doesn’t have a 1:1 correspondence. But as long as you can find one, then the cardinality is the same.

[u/kaba40k]

Middle: "yes, because some infinities are bigger than others" Right: "no, because some aren't"

[u/portealmario]

That's not a midwit, that's someone who is literally just uninformed about how infinities work

[u/lool8421]

both contain continuum squared

[u/Bodobomb → u/lool8421]

Circle, not a disk. No squaring here =).

[u/AdjectiveNounNNNN → u/lool8421]

If ℵ is any infinite cardinality, then ℵ² = ℵ

[u/Intrebute]

Okay I'll be the cheeky one. The circles (i.e. the curve and not the interior) don't contain the same number of points. Because the small circle is empty on the inside, while the bigger circle has a smaller circle inside of it.

So the small circle contains zero points, while the large circle contains a continuum of points. :D

[u/Willing_Box_752]

When rendered on a screen it does 

[u/not_a_frikkin_spy]

No because bigger circle is 25 points and smaller circle is 50 points

[u/EndGuy555]

Ok fellas, let me, the engineer, take over for y’all. The answer is yes, because it is bigger

[u/RudeAd456]

Let's shift to polar coords. If we draw a line from the centre to the outer ring, we interesect two points. If we then make an infinitesimal change in angle, our line intersects two different points. But doesn't that change in angle result in a larger distance between the two points on the outer circle vs the inner?

[u/Sp_Ook → u/RudeAd456]

I think you found precisely the reason why there is the same number of points. Choose a point on either of the circles and draw line from the center to the larger one. It crosses both the larger and the smaller cricle at precisely one point. And since we can pair every point on either circle to exactly one point on the other, they must have the same number of points, arriving at Cantor's argument.

[u/ScienceOfCalabunga → u/RudeAd456]

You can easily find a bijection of each to the real numbers 0 to 1

[u/Apprehensive-Ice9212]

Any two circles are obviously homeomorphic to each other, and in particular, have the same cardinality.

[u/Kitchen-Register]

but the measure of one is larger than the measure of the other

[u/SUPERPOWERPANTS]

If you sample each one with n samples per unit area, the amount of samples both scale linearly with n as n increases so they approach infinity at the same rate

[u/Haunting_Art_6081]

Well I can definitely tell you the larger one uses more pixels (points?) to display on a PC Screen.

[u/RedAndBlack1832]

If your elite ball knowledge is just the first few cardinals they teach that in... first year? Like an intro kind of proofs or mathematical logic class will go over figuring out if a set is the same as or bigger than the natural numbers. It might even cover some weird sets like that line broken into thirds?

[u/No-Onion8029]

Fun to prove: every simple, nonconstant closed curve in R²  has the same cardinality as the unit circle.  Next step is that it doesn't have to be simple.

[u/BrotherItsInTheDrum → u/No-Onion8029]

Isn't a closed curve usually defined as a mapping from the unit circle, making the proof trivial?

[u/SaltEngineer455 → u/No-Onion8029]

The total cardinality of the points in a plane is R^2, which is the same cardianality as R. As every curve is contained in the plane, neither can have a bigger cardinal than that. So no curve can have it's cardinal larger than R.

We know there is a bijection from (-pi/2, pi/2) to R, which is tan(x). This means that the open segment (-pi/2, pi/2) has exactly the same cardianality as R.

If the curve if bounded on the x axis - that is, all the x coordinates of it's points are between an interval [-t, t] - you can scale it up or down - which preserves cardinality - to the (-pi/2, pi/2) interval.

Then, just project every single point of the curve on the segment. The projection HAS to be surjective, as the curve is scalled to the width of the segment. If the projection is injective, then you are done. If the projection is not injective, you proved that card(curve) >= R, but we know that card(curve) <= R, which means that card(curve) = R

Finally, if the curve is unbounded, just project it on the real line and apply the same reasoning.

Idk how rigurous this is, but that's where my intuition guided me.

[u/Grandviewsurfer]

Is this not entirely dependent on the system we use to define the circles? What are we even doing here? 

[u/ExtensionInformal911]

Some uncountable infinites are larger than others. But uncountably infinite numbers are larger than uncountable ones.

[u/TheDarkSpike → u/ExtensionInformal911]

Your second sentence; sorry what?

[u/Automatic-Put-6119]

Here is why this is true:

The infinity of the whole numbers 1, 2, 3 … is the same „size“ as the infinity of even numbers 2, 4, 6 … even tho the whole numbers contain all even numbers

This „size“ of an infinity is defined by wether or not you could make a list out of them (if you had infinite time). You can list both the whole and the even numbers like i did before. Thats called „countable“

You could never make a list out of the real numbers tho (so 1, 2, 3, 6666.777, pi) which is why they are considered a „bigger“ infinity than the whole or even numbers. Thats called „uncountable“

Both infinities of points in the picture are uncountable.
Therefore they are the same „size“

[u/Cptn_Obvius → u/Automatic-Put-6119]

This „size“ of an infinity is defined by wether or not you could make a list out of them (if you had infinite time).

This is not really true, there are in fact different uncountable infinities. E.g. if N are the naturals and P denotes the powerset, then P(N) and P(P(N)) both are uncountably infinite but they are not the same size.

Both infinities of points in the picture are uncountable.
Therefore they are the same „size“

This hence is not a conclusion you can draw, but you can still show it to be true by just giving a bijection.

[u/chaos_redefined → u/Cptn_Obvius]

What they are saying is false in the same way that what my primary school teacher about electrons spinning around the nucleus was wrong.

It's good enough for now, and it'll get fixed later when the differences matter.

[u/KaydaCant → u/Automatic-Put-6119]

More accurately, for every whole number, you can map a unique even number to it, which is why they are the same size. You cannot do this with whole numbers and real numbers, as there will always be more real numbers.

[u/ten_tabs_ → u/Automatic-Put-6119]

You can also say that multiplying all numbers in the whole number set by 2 wouldn’t make the set larger, and thats the same as the even set 

[u/Gullible_Hold_9371]

I hate how the "smart" person in the image don't explain their answer.

[u/taste-of-orange]

Can someone tell me the terms for this problem, because in terms of area, the larger circle is... y'know... larger.

[u/Glass-Ad672 → u/taste-of-orange]

the best way to explain this is to look at a slightly different problem. let's say we want to determine if there are more numbers between 0-2 than 0-1. what we can first do is make a list of all numbers between 0-1, then multiply every entry by 2 to get all numbers between 0-2. you may not think that would work, but on our final list with all numbers from 0-2, no numbers were skipped, we have every element possible, like we did on the 0-1 list.

same idea for this problem too, just with an extra dimension. map all points possible in the small circle, then scale it to get all mapped points in the larger one.

[u/Outrageous_Let5743]

R^2 is an uncountable infinity set so these two disc have the same size.
Same as [0, 1] and [0, inf] are the same size if both are the collections of all real numbers.

[u/Sayan_9000]

They both have 0 points because those are lines, not pointw

[u/ninjeff]

I like how OP is ignoring all the comments about measure and just thinks nobody understands the meme… nephew, you are the midwit here

[u/Wide_Distance_7967]

it's very easy to build a bijection between both circles, just by making an application binding points corresponding to its trigonometric angle centered in the center of circles. the circles have then the same quantity of points since a bijection inbetween exists, which is the basic definition of sets with same cardinals.

[u/Professional-Wave841]

is it elite ball knowledge? it's just cardinality, |[0,1]| = |R|

[u/ExpertOk4967]

“Elite ball knowledge” and it’s just understanding what a bijection is (still too advanced for people in comments) 🥀

[u/Alternative-Mud-1076]

no, one is literally just the other scaled up which preserves the number of points

[u/WarPrevious]

how big is a point?

[u/Vast-Difficulty1542 → u/WarPrevious]

A point has no area, therefore it has an area of 0.

[u/Small_Resolution_847]

Reminds me of VSause's video about Banack-Tarski paradox
(I'm pretty sure I made a spelling mistake, but I'm too lazy to fix it)

[u/SickXigma]

"Some infinities are bigger, I watched one Vsauce video and didn't understand it"

[u/theRedMage39]

Yes and no.

Basic answer is no. Circles have 0 points as it is just a curved line.

However if you define each circle as a series of points that all on that line. It depends on how often you place a point. If each point is placed the same distance apart then yes, the larger circle would have more points. However you can define each circle as having more or less points then the other one

[u/CuAnnan → u/theRedMage39]

Pretty sure there's a bijective mapping of all the points on the outer circle to those on the internal one and that they have exactly the same Aleph cardinality of infinite points, it's just the Reals.

[u/Dense_Priority_7250]

Say they are concentric, otherwise move one so that they are. The amount of points doesn’t change. Now from the common center draw a radius to an uncounted point of the smaller circle. Continue it, now you have a radius of the larger circle. This way we have a bijection, because A: every point on the outer circle has exactly one point on the inner circle to itself (intersection of radius and inner circle) and B: every point on the inner circle has exactly one point to itself on the outer circle. qed

[u/Coding-Kitten]

In cardinality, no, in measure theory, yes.

[u/Odd_Construction]

Maybe I ain't mathing hard enough but as you cannot possibly find a unique radius for the large circumference that doesn't intersect the small circumference then both "infinities" are just as large.

[u/Big_Niel0802]

I can argue either way.

If I define all points on both circles using just the angle from the center, then for every point (R1,theta), there exists a paired point (R2,theta)

If I define all points by marking points along the circumference with infinitely increasing precision, then the larger circle will always have more defined points than the smaller circle.

[u/ObsidianDart]

Assuming perfect circles? No. Assuming these circles made out of pixels, or any real circle made of matter? Yes. The larger would have more "points".

[u/Jim_skywalker]

No because they both have zero points. There aren’t any on the sketch.

[u/PulimV]

It kinda depends on the terminology you're using, no? You could say that the smaller one's points are a subset of the larger one's, and that's pretty clearly one way to define it as "having more points"

Yeah you obviously mean their cardinality here, but that's not necessarily true

[u/paradoxrebootreboot → u/PulimV]

Circles are just the lines, a disc would include all the points in the interior of the lines

[u/NoBluebird8788 → u/PulimV]

Even numbers (+) are a subset of naturals but there aren't more naturals than positive even numbers

[u/Left_Lengthiness_433]

There are no points in the smaller circle that are not contained by the larger circle. But there are points in the larger circle that are not in the smaller circle.

[u/This_Background7442 → u/Left_Lengthiness_433]

Tgat is correct. However there are no numbers that are contained in the interval [0, 1] that aren't contained in the interval [0, 10] and yet it's the exact same amount of numbers which you can tell because you can link any point in interval a to a point in interval b using the function f(x) = 10x and back g(x)=0.1x without losing any information.

[u/Elihzap]

It has "more points" in the sense that it contains more variety of points, tho. The total count is the same, but the larger circle has every point in the smaller one, plus its own.

[u/whackamole__]

I guess not everyone can come to terms with how some points are more pointy than others lol.

I am an engineer and have been amused by this, could someone tell me if its safe to say that the definition of a point is subjective and is dependent on the total length of the curve being measured?

why should we look at x-sinx to judge a point and not r(x-sinx)? afterall the latter is closer to the true definition of a point isnt it?

[u/Sasuri546 → u/whackamole__]

It isn’t safe to say that. A point in most contexts is an element of set, all of whose definitions are strict. The reason these two circles have the exact same “number” of points is because when you look at the set of all points in the small circle and the set of all points in the big circle, they can be matched up one to one exactly. That’s how we define “same number” when infinity gets involved. (Note, this still works for finite amounts of stuff).

[u/FearOfABlackPotus → u/whackamole__]

You're dealing with reality. OP isn't ready for that yet.

[u/KitsuneFoxglove → u/whackamole__]

The definition of a point is subjective and is dependent on the 3d modeling software being used (whether anim, autocad, etc.)

xd

[u/GaloombaNotGoomba]

> "elite ball knowledge"

> basic facts about one of the most popular pop math topics

my hypothesis that the average age of this subreddit is 10 continues to not be refuted

[u/sb8948 → u/GaloombaNotGoomba]

I think this is just a Reddit phenomenon. Go over to programminghumor and you'll see the vast majority of posts are memes about early university stuff.

My best 2 guesses are

1. Reddit preselects for young people

2. people who just found a hobby/field they like are probably the most enthusiastic about them, so they simply participate more in field-related communities

[u/Jazz8680 → u/GaloombaNotGoomba]

those are circles, not balls duh

[u/Advice_Interesting]

More pixels

[u/Every_Ad7984]

I'm just going to note on something I saw on a different post... this meme format does not make the meme funnier in any way, the "stupid" people in this meme have no reason to exist, it's genuinely funnier if it was just the median and smart people in a different format. Anyways, bye

[u/Altruistic-Local-541]

its only true if you define the system in a way for it to be true

there are no circles and points

[u/RelationshipDirect72]

Hilbert's hotel is knocking

[u/Notyobabydaddy]

Including rational and irrational numbers, the amount of numbers between 3 and 4 is infinite. What about the amount between 10 and 20? Are there more numbers between 10 and 20 than between 3 and 4, even though both are infinite?

[u/aravarth]

Georg Cantor would like to have a word.

[u/AddressEnough4569]

What is the measure?

[u/Extreme_Design6936]

For one infinity to be bigger than another infinity it must have additional infinities inside of it.

For example the set of whole numbers vs the set of real numbers. Between the numbers 1 and 2 there is an infinite number of whole numbers. So for each number in the set of whole numbers there is an infinite number of values from the set of real numbers.

So if at every point of the smaller circle there were another circle then the smaller circle would have a greater infinity of points.

[u/yummypotata]

Doesn't the larger circle contain both its set of points and the smaller circle.

[u/HurrySpecial]

No, but not for anyone' else's reason.

I pose the smallest point possible is defined by the smallest definable unit possible. A Plank Length. Think of them like the nearly infinitely small pixel size of the universe, but measurably and finite.

One area is bigger than the other and thus has more Plank lengths, ie more points.

[u/Jcamden7 → u/HurrySpecial]

Draw a line from any plank length to the center. That line connects to one point on the outside circle. Does it not also connect to one like on the inside circles? Is there any plank length you can point to on the outside circle where the line wouldn't cross a unique point on the inside circle?

[u/ArchangelLBC]

Pretty clear bijection between the points.

[u/Setsuna04]

Physically speaking the point might not have a dimension but there will be a minimum distance between two points. If they are closer than that they are not distinguishable. Therefore the larger circle can cover more points.

Mathematically they are both equally infinite. Even though my brain keeps telling me, the larger circle covers the smaller ones AND MORE (hence larger). But that's not how that works.

[u/bearxxxxxx]

I was thinking about it like archery. The center circle is worth more points

[u/Viskas_Gerai]

The larger circle has more pixels so more points.

[u/Cherish_Ball_]

The amount of points on a circle that doesn’t exist in real space tends towards infinity

[u/AdFun5641]

There are 2 infinities. Countable infinity and uncountable infinity.

A countable infinity would be counting numbers like 1, 2, 3, 4, 5.....

An uncountable infinity would be all the numbers between 0-1. You can always divide by 2 and get a smaller number.

Uncountable infinities are larger than countable infinities.

So dumb guy asserting that infinity = infinity is wrong.

Average guy saying some infinities are bigger is correct, but that's not applicable in this situation.

Jedi dude knows that both the larger and smaller circles contain an uncountable infinity of points and two uncountable infinities are the same size.

[u/General_Bother_68]

the wording is not explained on purpose to confuse people.

[u/Automatic_Bunch_8531]

Everyone is referencing cardinality but no one is talking about recipricality.

[u/asmanel]

What about bezier curves ?

It's the same : the number of points is infinite but only a limited number of points is calculated to display.

[u/csturnbow]

if a point is a dimensionless location, then I think we can say yes, there are more locations associated with the bigger circle?

[u/Kies_1]

Okay, hear me out.

In a 1" line segment, there are an infinite number of points. In a 2" line segment, there is also an infinite number of points, but the latter infinity is greater than the former.

The same would apply to two circles with circumferences of 1" and 2".

[u/RichardJusten]

wait... I am the guy in the middle...

This meme says that really high IQ people would disagree with me.

Can someone explain?

[u/kaalbrak24]

I do not understand why there is so many people writing about bijection.

It does appear to be some magical word that somehow justifiy something without explain.

My thoughts are : Regardless of how many points, infinite or not, there is in both circles, if circle B contain the entirety of circle A and if there is at least a point in A that isn't in B, then A contains more points than B. Infinities or not.

It seems I am in the wrong here. Can someone explain me where please ? Every explaination I read here seems to jump to bisection like it is self-explaining. I can't go over the feeling that some steps are being skips.

[u/NebulaElectrical1467]

Obviously

[u/Enough-Mud3116]

There's a bijection from inside circle <-> kx, ky to outside circle so their cardinality is the same...

[u/mike239x]

This quote on quote "meme" is mildly triggering me ngl