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u/Ashamed_Association8 4d ago
Is everybody being blind here? As far as i can see neither of these circles has any points inside them.
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u/Accomplished_Bad_487 Transcendental 4d ago
The bigger circle has actually infinitely many points inside of it, the smaller doesnt have any tho ._.
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u/gmailster 3d ago
circle has infinite points? i thought circle is not points, only pointless smooth surface
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u/patientpedestrian 3d ago
Like a Ken doll's crotch!
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u/gmailster 2d ago
what
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u/Environmental-Try736 4d ago edited 3d ago
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
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u/AdMuch1724 3d ago
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
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u/LGKouglof 4d ago
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u/Ex_Federa 4d ago
Some infinities are bigger than others, but not these ones.
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u/Mr_kalas22 Real Algebraic 4d ago
More like some infinities are bigger than others unless we can find a bijection between them
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u/belabacsijolvan 4d ago
*some infinites are bigger than others if there cant be a bijection between them
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u/Appropriate-Sky-7798 3d ago
... if they're not equal and we accept the axiom of Choice
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u/belabacsijolvan 3d ago
if they are equal then the identity map is a suitable bijection.
the axiom of choice is a good pedancy tho
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u/djta94 3d ago edited 3d ago
EDIT: and there is at least one injection between them
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u/belabacsijolvan 3d ago
thats not true. y=x is a non-surjective injection from [0,1] to [0,2].
yet their cardinality is the same as shown by y=2x bijection.5
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u/Hapcoool 2d ago
*infinities are bigger or smaller than others if there cant be a bijection between them
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u/creeper6530 Engineering 4d ago
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?
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u/TeraFlint 4d ago
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. >_>
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u/Miserable-Repair-191 4d ago
Doesn't a black page contain every possible proof ov anything drawn on it, including this one?
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u/Difficult-Nobody-453 4d ago
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
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u/creeper6530 Engineering 4d ago
I mean, sure, bijectivity makes inverse functions possible so you can just swap the words "inside" and "outside"
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u/Old_Patience_4001 4d ago
I think what he's saying is that someone might think the larger circle has more points than the smaller circle, so it would be a better explanation for them to say every point on the larger circle has a corresponding point on the smaller circle rather than the other way round.
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u/XNo_Notes 4d ago
It seems to me that would be equivalent to drawing a third circle and doing what creeper6530 said.
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u/AsleepDeparture5710 3d ago edited 3d ago
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.
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u/gavlna 4d ago
how is this bijection? There are (if we are in a R² anyways) infinitely many points on such a line
nvm, circle aparently is just the outer line, not the inside
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u/creeper6530 Engineering 4d ago
Formally, the set of all points with distance to centre less than or equal to
ris termed a disc, not a circle. A circle is just all points with distance to centre equal tor.In other words, disc is the region of the plane bounded by a circle.
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u/Layton_Jr Mathematics 4d ago
Even if it were a disc, a scaling transformation is always a bijection unless the scaling factor is 0
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u/HephMelter 4d ago
In any case, it is always possible to creare a bijection from any portion of the plane to any other as long as both have a non-zero Lebesgue measure (you can squish a half-plane down to a square of arbitrary area, and you can stretch one such square to any form of your liking) so even the ring and the disc could be put into bijection
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u/thussy-obliterator 3d ago
Somewhat informally
for all r in R where r > 0 there exists a set C_r where C_r is a subset of C such that for all x in R where x ≥ 0 and x < 2π r * e^(ix) is in C_rThe cardinality of set C_r is independent of r and is dependent on the set x ≥ 0 and x < 2π
Therefore all C_rs are of an equivalent cardinality
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u/kdeberk 4d ago
Some infinities are bigger than others, some infinities' mothers are bigger than other infinities' mothers
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u/somedave 4d ago
I mean it uses more pixels on the screen
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u/Monoid-Confessor 4d ago
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.
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u/Hungry_Painter_9113 4d ago
I mean, you don't even need to say that, assuming both sets are subset of R2, say smaller circle is M1 and bigger one is M2 |M1|<=|M2|<=|R2|, you can easily prove that x2+y2<=r12 has an uncountable infinite cardinality solution set hence |M2|=|M1|
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u/Technical-Ad-7008 Mathematics 4d ago
Your proof looks more like a cannon to a fly than the other guy though. Hahah
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u/ThirstyOutward 4d ago
You don't even need to do that, you could do something even more complicated lmao
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u/Layton_Jr Mathematics 4d ago
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 xa+yb (you can even write z^(c + d) to get zc + d
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u/Liko81 3d ago
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.
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u/AdequatlyAdequate 3d ago
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"
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u/klimmesil 3d ago
But feathers are lighter than steel
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u/Waffle-Gaming 3d ago
yes, but they're both a kilogram
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u/ScuffedMobilePlayer 3d ago
Yes but the feathers also contain the weight of what you did to that bird.
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u/GeneralParticular663 3d ago
yes but i shoot blanks after the vasectomy
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u/Mordret10 3d ago
Because people assume infinity to be something growing, rather than something static. And f(X)=20x is growing faster than f(X)=X
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u/MartyMcBird 3d ago
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 :(
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u/cornking 3d ago
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
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u/psychicesp 3d ago
Lol, I wouldn't. You overestimate what people are willing to regurgitate without understanding
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u/Ka-Jin 3d ago
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.
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u/DaleGribble88 3d ago
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.
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u/CogentCogitations 3d ago
And yet you can also map every point in the smaller circle to the exact same point in the larger circle, so what would the left over points in the larger circle map to in the smaller circle?
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u/frothyQratos 3d ago
If I remember correctly, for uncountable infinities you can’t employ this logic, such as in Cantor’s diagonalization. I believe you could always find a new point that doesn’t exist in your mapping
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u/donach69 3d ago
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
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u/GlitteringBlood2005 3d ago
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
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u/FormerlyPie 3d ago
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
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u/Drag0n_TamerAK 3d ago
Believe it or not most people are kinda stupid and just parrot phrase other people have said until they loose all meaning
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u/the_horse_gamer 3d ago
people who say "some infinities are bigger" just watched a vsauce video on their second monitor
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u/sconosciuto_io 4d ago
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)
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u/miguescout 4d ago
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
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u/catman__321 4d ago
Technically not a sum but a measure function designed with the structure of the space you're taking the area from in mind
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u/somethingX Physics 4d ago
That's more units of measurement than the formulas. Every unit of distance is infinity times 0 with extra steps
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u/Downtown_Finance_661 4d ago
Bro, area is number of points times area of point. Turn in your degrees.
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u/BIGBADLENIN 3d ago edited 3d ago
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)
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u/grottoe24 3d ago
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.
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u/Cheltorius 3d ago
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.'
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u/WineSauces 3d ago edited 3d ago
Yeah exactly, in the concrete sense we can suppose a finite number of point particles within each defined space, but additionally, in a concrete sense we can suppose an infinite number of positions for those point particles.
The question of course is too vague of the system to actually force any definitive answer.
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u/paradizingmania 3d ago
I mean sure they’re using two different definitions of “size” but it’s not just semantics and one of the definitions is correct while the other understanding simply isn’t how math works.
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u/j0shred1 4d ago
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?
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u/lifeistrulyawesome 4d ago
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"
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u/Muroid 4d ago
The circles themselves are not the same size, but the set of all points contained within each circle is the same size.
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u/lifeistrulyawesome 4d ago
The circumferences (the borders) have the same cardinality, but different lengths
The circles (all the points inside) have the same cardinality but different areas
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u/Bth8 4d ago
The border is the circle. Circumference is the length of a circle. The interior is a disc.
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u/lifeistrulyawesome 4d ago
I have heard different usages. As a kid, I was taught in school that the circle is what you call the disc and the circumference is what you call the border. But if someone used the terms differently, I wouldn't mind. I don't care about what your preferred terms are, as s long as we understand each other.
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u/Bakuryu91 4d ago
The larger circle C1 is the set defined by {x1,y1 where x1²+y1²=r, x1, y1 in R}.
The smaller circle C2 is the set defined by {x2,y2 where x2²+y2²=r, x2, y2 in R}.
I don't think they're the same size, am I missing something?
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u/ChemBro93 3d ago
They’re the same number of points since they have the same quantifiable number of points.
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u/pmmeuranimetiddies 4d ago
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?
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u/Kalmoran 4d ago
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.
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u/pmmeuranimetiddies 4d ago
Yeah, that's more or less what my physical intuition was i just don't know if that's considered topographically rigorous proof
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u/WurmcoilEngine11 4d ago
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
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u/nakedascus 4d ago
but why are you restricted to lines that radiate from the center?
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u/portealmario 4d ago
Because that makes the proof easier and clearer, while guaranteeing that every point of both circles in accounted for
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u/BrotherItsInTheDrum 4d ago
I think naive people start in the middle of this chart, not on the left.
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u/MaxTHC Whole 3d ago
Nah, naïve people actually look at the top image and say "I don't see any points"
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u/HorsefaceWithNoName 4d ago
But how many angels can dance on the head of a pin that fits within each circle?
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u/Neither_Review_3470 Mathematics 4d ago
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
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u/Legal_Weekend_7981 4d ago
Regardless of interpretation, the number of points is the same.
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u/boterkoeken Average #🧐-theory-🧐 user 4d ago
Question unclear. I accidentally proved the continuum hypothesis.
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u/doctornoodlearms 4d ago
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..
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u/tigerthetabbyy 3d ago
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)
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u/_Denizen_ 3d ago
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.
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u/tmbmad 3d ago edited 3d ago
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.
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u/pikachu_sashimi 3d ago
A philosopher who declines to answer because of not constraining the question to a school of thought should stop being a philosopher.
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u/_Kian_7567 4d ago
You can draw lines between the circles such that every point one matches one one on the other so it’s the same amount
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u/portealmario 4d ago
That's not a midwit, that's someone who is literally just uninformed about how infinities work
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u/Intrebute 4d ago
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
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u/not_a_frikkin_spy 4d ago
No because bigger circle is 25 points and smaller circle is 50 points
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u/EndGuy555 4d ago
Ok fellas, let me, the engineer, take over for y’all. The answer is yes, because it is bigger
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u/RudeAd456 3d ago
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?
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u/Sp_Ook 3d ago
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.
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u/Apprehensive-Ice9212 4d ago
Any two circles are obviously homeomorphic to each other, and in particular, have the same cardinality.
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u/pmmeuranimetiddies 4d ago
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?
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u/LitespeedClassic 3d ago
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.
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u/anooblol 2d ago
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.
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u/Kitchen-Register 4d ago
but the measure of one is larger than the measure of the other
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u/SUPERPOWERPANTS 4d ago
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
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u/Haunting_Art_6081 3d ago
Well I can definitely tell you the larger one uses more pixels (points?) to display on a PC Screen.
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u/RedAndBlack1832 3d ago
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?
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u/No-Onion8029 4d ago
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.
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u/BrotherItsInTheDrum 4d ago
Isn't a closed curve usually defined as a mapping from the unit circle, making the proof trivial?
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u/No-Onion8029 4d ago
For the simple case, I think you should at least mention Cantor–Schröder–Bernstein. For the general case you need continuity, the IVT, a parameterization, and C-S-B.
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u/SaltEngineer455 4d ago
The total cardinality of the points in a plane is R2, 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.
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u/Grandviewsurfer 4d ago edited 3d ago
Is this not entirely dependent on the system we use to define the circles? What are we even doing here?
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u/ExtensionInformal911 4d ago
Some uncountable infinites are larger than others. But uncountably infinite numbers are larger than uncountable ones.
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u/Automatic-Put-6119 3d ago
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“
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u/Cptn_Obvius 3d ago
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.
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u/chaos_redefined 3d ago
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.
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u/KaydaCant 3d ago
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.
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u/ten_tabs_ 3d ago
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
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u/Outrageous_Let5743 4d ago
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.
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u/Wide_Distance_7967 4d ago
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.
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u/ExpertOk4967 4d ago
“Elite ball knowledge” and it’s just understanding what a bijection is (still too advanced for people in comments) 🥀
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u/Alternative-Mud-1076 4d ago
no, one is literally just the other scaled up which preserves the number of points
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u/Small_Resolution_847 3d ago
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)
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u/SickXigma 3d ago
"Some infinities are bigger, I watched one Vsauce video and didn't understand it"
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u/theRedMage39 3d ago
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
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u/CuAnnan 3d ago
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.
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u/Dense_Priority_7250 4d ago
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
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u/Odd_Construction 4d ago
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.
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u/Big_Niel0802 4d ago edited 4d ago
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.
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u/Ma4r 4d ago edited 4d ago
No, it is SUFFICIENT that a bijection exists between two sets to call their cardinalities equal. So just because of the fact that you found a way to define a point on both circles you can no longer say one one of them has more points (according to set theory at least)
The way you could differentiate them is by using measure theory, but if we're using measure theory we can't exactly speak about the number of points (or cardinality rather) in each circles. But measure theorsists aren't real people anyways so that's not a concern
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u/ObsidianDart 4d ago
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".
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u/PulimV 4d ago
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
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u/paradoxrebootreboot 4d ago edited 4d ago
Circles are just the lines, a disc would include all the points in the interior of the lines
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u/aspiring_scientist97 4d ago
Two finite lines, and infinite ones, still have the same cardinality number of points no?
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u/PulimV 4d ago
Oh
Oops lmfao
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u/paradoxrebootreboot 4d ago
It's ok I made this mistake until years after I finished my degree lmfao
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u/NoBluebird8788 4d ago
Even numbers (+) are a subset of naturals but there aren't more naturals than positive even numbers
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u/_404_LogicNotFound_ 4d ago
The amount of people disagreeing with this is just diabolical...
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u/your_best_1 4d ago
Infinite is tough, random is too
Really hard for people to get their heads around.
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