A line doesn't have the same number of points as a square

[Anonymous No.14791099]

https://mindmatters.ai/2022/07/in-infinity-lines-and-squares-have-an-equal-number-of-points/
This is the dumbest thing I've ever seen

[Anonymous No.14791113]

What exactly is giving you trouble, OP?

[Anonymous No.14791130]

>>14791113
A square is massively bigger. If the line is overlaid on top of the square then throwing a dart at it should have an equal chance of hitting the line vs the white part if they have the same number of points, but that is nonsense.

[Anonymous No.14791314]

>>14791130
>a square is massively bigger
Bigger in what sense?

[Anonymous No.14791321]

>>14791130
>conflating measures with cardinality

You are comparing two entirely different things. A line has 0 measure in 2D. A line has the same cardinality (of points) as a square. These are compatible statements.

[Anonymous No.14791587]

>>14791314
More points. Does the dart analogy not make sense?

>>14791321
But more than 1 line can fit in the square. There's a horizontal line at .5, another one at 0.75,another in infinity locations.... so it obviously can't have the same cardinality/ same number of points. The "proof" in the link only used irrational numbers to correspond points between a line and square btw. That dumb "trick" he uses doesn't work for numbers with repeating zeros. ie point 0.50... on the line has no specific corresponding point within the square.
I mean hell, even the perimeter of the square does not have same cardinality with the line. It has 4x as many points. Now include the rest of the points in the square. That is as intuitive as you can get.

[Anonymous No.14791614]

This is either a brilliant troll post or someone is getting filtered by a first semester lemma

[Anonymous No.14791620]

>>14791099
Infinity is like floating point number overflow in CS, it corrupts every single equation that interacts with such number afterwards

It surprises me mathematicians waste time on infinity problems, must be hobby-only thing

[Anonymous No.14791741]

>>14791099
You don't understand math, retard.

[Anonymous No.14791766]

>>14791587
The dart analogy is wrong because measure has nothing to do with number of points. [0, 0.1] and [0.5, 1] have the same number of points, but if you select a number between 0 and 1, you have 5x probability of being in the second interval.

As for your second point, how many lines you need is irrelevant. You need two rays to tile the real number line, but they have the same cardinality. You need an infinite number of squares to tile the plane, but they have the same cardinality. The trick with zeros works perfectly fine if you include the boundary of the square, but even if you don't, you can avoid it via an offset by a bijective function that never attains a value it tends towards (like arctan which also can be used to show that a line and a line segment have the same number of points.

>I mean hell, even the perimeter of the square does not have same cardinality with the line. It has 4x as many points. Now include the rest of the points in the square. That is as intuitive as you can get.

Entirely incorrect. I suggest you read up on measure theory.

[Anonymous No.14792278]

>know nothing about maths
>get confused and angry
>make a retarded post on 4cheddit
sad.. many such cases

[Anonymous No.14793296]

>>14791766
>[0, 0.1] and [0.5, 1] have the same number of points
But every single point that fills up the first interval only fills 1/5th the 2nd. They can't have the same number of points outside of made up a priori definitions. ie the midway point of the 1st only maps to the midway point of 1/5th the span of the 2nd interval.
>You need two rays to tile the real number line, but they have the same cardinality
We're talking about a line segment of distance 1, not the entire real number line.
>The trick with zeros works perfectly fine
I think you misunderstood. He made up a trick. The zeros dilemma defeats it. All he proved was a 1 to 1 mapping of irrational numbers between a line of 1 and a unit square
>you can avoid it via an offset by a bijective function that never attains a value it tends towards
This is just saying there's the same number of points if we ignore some points on the square
>Entirely incorrect. I suggest you read up on measure theory.
This is prefaced on your 5x claim above that is clearly deficient
>>14791741
>that pic
Good bait
>>14792278
who's angry?

[Anonymous No.14793369]

>>14793296
>They can't have the same number of points
I can make a bijective function from [0,0.1] to [0.5,1], which means there are the same number of points

[Anonymous No.14793431]

>>14793296
Let y = 5x
For all x in the interval [0, 1] there's a y found in the interval [0, 5].
For all y in the interval [0, 5] there's a corresponding x in the interval [0, 1].
There exists no such y for which there's no corresponding x.
The have the same cardinality and thus there's as many reals in [0, 1] as there are in [0, 5].
Unless of course you have an example of such a bijection of different real number intervals where one of them has no matching element from the other.

[Anonymous No.14793712]

>>14793369
That is illogical. You're saying one segment's points can be mapped to the points of 5 other separate segments of itself
>>14793431
>For all x in the interval [0, 1] there's a y found in the interval [0, 5].
Sure, but there's also more Y left over once you have mapped all of X. If I have 10 marbles and you have 50, my first 10 can be mapped to the first 10 of yours, but you have some left over. If we increase our sets proportionately there is no amount of marbles I can have that will ever map to all of yours.
btw I'm not saying "10*infinity < 50*infinity" I'm saying they can't be mapped to each other. Think of the Grand Hotel paradox but with two hotels, one with 10 floors and the other with 50. You can map the first 10 floors of the larger to the first 10 of the smaller (make them roomates lol), but that is it. The other 40 floors have infinity left over who have nobody to be roomates with in the smaller hotel.
>For all y in the interval [0, 5] there's a corresponding x in the interval [0, 1].
Nope. That's just working backwards from above. Your 50 can't all be mapped to my 10
>The have the same cardinality and thus there's as many reals in [0, 1] as there are in [0, 5].
You're pressing the "I believe" button to reach this conclusion.

[Anonymous No.14793777]

>>14793712
>there's also more Y left over once you have mapped all of X
Nonsensical statement. What does it even mean to map all of them such that there's left overs?
>If I have 10 marbles
Why are you bringing finite sets into this now? I thought we were talking about cardinality of infinite sets.
>btw I'm not saying "10*infinity < 50*infinity"
Good
>I'm saying they can't be mapped to each other
Wrong. I gave you a perfectly valid function that maps one real number line to another.
>Think of the Grand Hotel paradox
You are trying too hard to use big words and concepts but you don't even know what cardinality means.
>Your 50 can't all be mapped to my 10
I invite you to give me even a single example of an element of the [0, 5] set that doesn't map to [0, 1].
Do you also think that linear functions are discontinuous? Because this is kind of what you are implying.

[Anonymous No.14793800]

>>14791614
>if it is taught to midwits it must be true

[Anonymous No.14793919]

>>14793777
>What does it even mean to map all of them such that there's left overs?
Refer to the marble example, then extend it to infinity like in the hotel example that you seem to have skipped over...
>Why are you bringing finite sets into this now? I thought we were talking about cardinality of infinite sets.
It's called a primer to introduce a subsequent more advanced concept.
>Wrong. I gave you a perfectly valid function that maps one real number line to another.
No, you gave a valid function and an empty claim that one line is mapped to another.
>You are trying too hard to use big words and concepts but you don't even know what cardinality means.
Why do you complain when I introduce a finite set example, but when I expound on that to turn it into an infinite set example, you ignore it?
>I invite you to give me even a single example of an element of the [0, 5] set that doesn't map to [0, 1].
Easy. The very first number that comes after 1. There is no mathematical representation for this number. Hence you can't mathematically prove if it's within the other set, you can only claim it is. Ergo you can't map it. You're just pressing the "I believe" button like I said.
>Do you also think that linear functions are discontinuous? Because this is kind of what you are implying.
No not at all. But you do bring up a good idea. Say we point-by-point split up the y=.5 horizontal line in the unit square I mentioned above. This makes it discontinuous obviously. Now move each point by a random Y value between -0.5 and +0.5 placing it somewhere else in the square... according to your thinking you must affirm all points in the square will be covered because "all are mapped" to the line's points. That is nonsense.

[Anonymous No.14793930]

>>14793919
>Easy. The very first number that comes after 1.
There's no such number.

[Anonymous No.14793938]

>>14793919
>Easy. The very first number that comes after 1. There is no mathematical representation for this number. Hence you can't mathematically prove if it's within the other set, you can only claim it is.
Congrats. You just discovered the real number line. There is no such thing as "number that comes after 1" under reals.
So your proof that a function doesn't map one set of real numbers to another is that when you use a non-real argument it doesn't work? Genius. Btw, you must be over 18 to post.
>You're just pressing the "I believe" button like I said.
Being that guy in middle school who counters the math teacher with "umm actually 0.999... never reaches 1" is not cool. You're just being a contrarian and you don't have any actual points.

[Anonymous No.14793944]

>>14793919
>No not at all.
That is exactly what that implies. All function that don't have the form [math]f(x)=x+a[/math] are discontinuous according to your logic. I guess differential calculus is fake now. We can't even perform any differentiation on any curve anymore. We can all go home now.

[Anonymous No.14794030]

>>14791587
>But more than 1 line can fit in the square.
Forget lines and squares, they are just needlessly complicating things. First make sure you are comfortable with integers. The even integers has the same cardinality as the integers, despite the even integers being a proper subset of the integers.

[Anonymous No.14794154]

>>14791587
>More points.
But there are infinite points

[Anonymous No.14794192]

>>14794154
when infinities are involved, you have to resort to forming bijections between things to count them.
there are infinite integers, and i can enumerate them in a list.
there are infinite real numbers, and i can't enumerate them in a list.
it is discovered that in some sense there are more reals than integers.
and there are larger sets than the reals. for instance, the power set of reals (the set of all subsets of a set) is known to have cardinality larger than the reals (the power set of the reals is "bigger" than the reals). you can keep constructing "bigger" sets this way.

[Anonymous No.14794195]

>>14791587
>0.50...
[math]0.5\bar{0}=0.5[/math]. Even the [math]0.999...\ne 1[/math] posters would agree with that.

[Anonymous No.14794196]

>>14794192
GROUNDBREAKING.


Now do it on paper

[Anonymous No.14795167]

>>14793930
>There's no such number.
It's an infinitesimal number
>>14793938
>There is no such thing as "number that comes after 1" under reals.
an infinitesimal number does not exist under the reals.
If you want to limit the discussion to reals then fine. Ignore the next number(s) that comes after 1, what is the first real number that comes after 1 that we can define?
I suppose I should have said there's no universally accepted way to represent the "very first number that comes after 1"
>Being that guy in middle school who counters the math teacher with "umm actually 0.999... never reaches 1" is not cool
Strawman.
>>14793944
>That is exactly what that implies. All function that don't have the form f(x)=x+a are discontinuous according to your logic
Did you make a typo? That is a linear function and you said I imply linear functions are discontinuous but are now saying only non linear functions are implied to be discontinuous..
>>14794154
Some infinities are larger than others
>>14794195
I shouldn't have put "..." after the 5 that was redundant of me.
>>14794030
>Forget lines and squares
Uh no? You're saying ignore my argument and only listen to other people's arguments
>they are just needlessly complicating things
They're very simple to me I don't know what you're talking about.

[frenanon No.14795347]

>>14793712
I get what you're saying, and I agree with you.

That's why Infinity +/- Numbers should be treated as imaginary or hypothetical numbers.

If you are presented with 3 Programs to run on your computer, and 2 of those programs cause a crash through infinitesimal recursive hangs, then; then that is a greater chance of "infinity", as opposed to just one of those programs having an infinity glitch.

[frenanon No.14795352]

>>14793777
>I invite you to give me even a single example of an element of the [0, 5] set that doesn't map to [0, 1].
How about:
>[0+1, 1]

[frenanon No.14795376]

>>14794192
>when infinities are involved, you have to resort to forming bijections between things to count them.
Just looked up with a bijection is, but yeah, that's kind of what I was thinking as well. I think that's what I was trying to say in >>14795347 , but not sure.

If you are comparing the possible numbers of a circle's radius vs it's circumference; there will, by definition be more points using the line of the circumference, vs the line of the radius.

[Anonymous No.14796010]

>>14795167
>Did you make a typo? That is a linear function and you said I imply linear functions are discontinuous but are now saying only non linear functions are implied to be discontinuous..
If you actually read what I wrote you'd see what I mean. By your logic only functions of the form [math]f(x)=x+a[/math] are continuous. That is not the general form of a linear function if you didn't notice.

[Anonymous No.14796019]

>>14795376
>by definition
What definition?

[Anonymous No.14796290]

>>14795347
>I get what you're saying, and I agree with you.
Great. I'm just trying to make a constructive thread.
This topic has been on my mind recently from people saying the universe is expanding and casually overlooking the implication that more points are being created and added to the universe from nothing/out of nowhere as the universe is expanding. That doesn't sit well with me.
>That's why Infinity +/- Numbers should be treated as imaginary or hypothetical numbers.
They're certainly not conventionally useful. Though, imaginary numbers originally were laughed at as being useless.

>>14796010
>If you actually read what I wrote you'd see what I mean
I wasn't certain if you made a typo
>By your logic only functions of the form f(x)=x+a are continuous
No, they're the only ones that can have a 1 to 1 X to Y mapping according to the schema in your original example.
If we could zoom in "all the way" and look at the very infinitesimals themselves I'm saying it might conceptually look something like pic related (a very convenient online pic I found that illustrates my point). It's obvious X and Y only maps 1 to 1 for the graph labelled 1/2 (it's not mathematically rigorous so don't get too hung up on it)
>That is not the general form of a linear function if you didn't notice
of course noticed that. I wasn't sure if it's exactly what you meant or it was a typo.

[Anonymous No.14796300]

>>14795347
>I get what you're saying, and I agree with you.
Great. I'm just trying to make a constructive thread.
This topic has been on my mind recently from people saying the universe is expanding and casually overlooking the implication that more points are being created and added to the universe from nothing/out of nowhere as the universe is expanding. That doesn't sit well with me.
>That's why Infinity +/- Numbers should be treated as imaginary or hypothetical numbers.
They're certainly not conventionally useful. Though, imaginary numbers originally were laughed at as being useless.

>>14796010
>If you actually read what I wrote you'd see what I mean
I wasn't certain if you made a typo
>By your logic only functions of the form f(x)=x+a are continuous
No, they're the only ones that can have a 1 to 1 X to Y mapping according to the schema in your original example.
If we could zoom in "all the way" and look at the very infinitesimals themselves I'm saying it might conceptually look something like pic related (a very convenient online pic I found that illustrates my point). It's obvious X and Y pixels only maps 1 to 1 for the graph labelled 1/2 (it's not mathematically rigorous so don't get too hung up on it)
>That is not the general form of a linear function if you didn't notice
of course noticed that. I wasn't sure if it's exactly what you meant or it was a typo.

[Anonymous No.14796315]

>>14796300
>>14796010
Pic related is a little better visualization

[Anonymous No.14796320]

>>14791099
they do not have a number of points
it is uncountably infinite

[Anonymous No.14796325]

>>14791099
>>14791130
>>14791587
>>14793296
>>14793712
>>14796300
>>14796315
This is bait. There's no other explanation.

[Anonymous No.14796342]

>>14793712
>Sure, but there's also more Y left over once you have mapped all of X.
Take an interval like [0, 10] and divide every element by 2. You now have an interval [0, 5] which is indistinguishable from any other [0, 5] interval. They both contain the same elements so they are the same. And since we obtained it by dividing every element of [0, 10] (without adding or removing any) we know that they have the same number of elements.
I don't understand what the issue here is.

[frenanon No.14796374]

>>14796019
>What definition?
Circumference = 2πR
So by definition, if you were to take Line of a Circle's Radius, and then took the Circumference of that same Circle; the Circumference will inherently have a larger set of numbers.

If you have a magic bag with every possible number in existence, but some numbers are repeated multiple times; those repeat numbers have a higher chance at being drawn, despite there being Infinity numbers.

[Anonymous No.14796453]

>>14791099
OP, there is a bijection between the number of points, but measures such that the measure of both objects is non-zero (1 and 2 dimensional Lebesgue measure, for example), give the square's interior infinitely more measure than the line's. I can explain this rigorously if you want to.

[Anonymous No.14796475]

>>14793938
I would add that you can leverage the well-ordering theorem if you accept axiom of choice to show that there is a number that comes directly after 1, it just won't be the same order that he's thinking of.

[Anonymous No.14797042]

>>14796374
Scaling a line doesn't change the number of points it has.

[Sage No.14797062]

>>14791099
In my understanding infinity isn't a number, it's a concept to help explain how our equations react when approaching asymptotes. Of course a line doesn't have as many points as a square. They can both have "infinite" points in whatever structure you give it but that isn't an actual measure of anything. If you applied the same scale of metric to each structure you would have more points in the square every time.

[Anonymous No.14797087]

>>14797062
>Of course a line doesn't have as many points as a square.
[math]\mathbb{R}^2[/math] and [math]\mathbb{R}[/math] have the same cardinality. You can also construct a bijection [math]f: \mathbb{R} \rightarrow \mathbb{R}^\mathbb{N}[/math] which means that a line has as many points as a hypercube as well.

[frenanon No.14797307]

>>14797042
>Scaling a line doesn't change the number of points it has.
I disagree.
If you have a 6-point object that is fractally oriented, you will have "infinite" points.
But some of those "points" might be more/less readily accessible depending on the "angle(dimension)" you are viewing the iterated aggregation.

picrel is a bad example of what I'm trying to say, but somewhat relevant

[frenanon No.14797330]

>>14797307
Or if you consider different Conic Sections.
You have identical Circles, but if they are oriented such that they are larger/smaller, you could have points that are exclusive to one line vs the other.

[frenanon No.14797342]

>>14797087
>R2R2 and RR have the same cardinality
1 =/= -1

[Anonymous No.14797425]

>>14797307
>I disagree.
Too bad. You can scale a line just by linearly transforming each point. If you multiply each element of a set by a constant, you get a different set representing a line of different length but it no points were removed or added so they can't have a different number of points.

[Anonymous No.14797434]

>>14791099
It makes sense to me OP, would you argue that there's more space in 2D than 1D space, or less space in 2D, than in 3D? It's nonsensical to do so, and could be left at that.

Their other one claiming that the universe "began" is even more heinous.

[Anonymous No.14798241]

>>14791099
op, infinity is infinity. that's the thing about it.
there's just as many 'possible numbers' between 0 and 0.1 as there are between 1 and 100, that's all your picture is saying.
the reason for this is obvious. no matter how many numbers you have between 0 and 0.1, you can get twice that amount by multiplying it by 0.1, and have it still be in that range.

this means that the 'amount of numbers' is actually independent of the range. if [0,0.1] = [0,0.1]*2 by way of multiplying by 0.1, and so [0,0.1] = [0,0.2], which also doubles the possible numbers. In fact, similarly you can show this is true for any other multiplication, and for any start and end point. [a,b] = [c,d].

they're all equal, as long as they're in the reals, because the "number of numbers" in a range is equal.

[Anonymous No.14798243]

>>14795167
>It's an infinitesimal number

ok, my bad, i posted. didn't realize this was a bait thread, sorry for bumping.

[frenanon No.14799265]

>>14797425
>You can scale a line just by linearly transforming each point.
Wrong.
>If you multiply each element of a set by a constant, you get a different set representing a line of different length but it no points were removed or added so they can't have a different number of points.
No, this isn't true.

>Be Circle
>Have 2 Lines, Radius, and Circumference
>Shrink Radius down to the smallest number in existence
>The Circumference shrinks down as it scales with the Radius
>Oh wait; the Circumference literally can NOT shrink down to the size of the Radius, because of [2πr]
>To do so, you would have to reconfigure the properties of the Line(s) that make the Circle, Or;
>Reorient the Circle's Plane/Angle of Observation, so as to produce an answer that satisfies the solution

>>14797434
>would you argue that there's more space in 2D than 1D space, or less space in 2D, than in 3D?
There's exponentially more "Space" between 3D v 2D, than there is from 2D v 1D

[Anonymous No.14799315]

>>14799265
You didn't actually explain why you can't do it. You just asserted that it's impossible.
Also, I doubt anyone would accept that you can have fractional or irrational pieces of a point therefore either the radius or the circumference cannot exist under your logic.

[frenanon No.14799354]

>>14799315
>Let me tell you about infinity
>No, you can't use irrational fractions, because those are non-ending infinites, and infinites aren't real
>You can't use numbers that are naturally infinite, you have to choose a Whole number
>But this Finite Line has an infinite amount of points on it, so that's ok
Seriously dude? Do you not see how ridiculous that is?
>either the radius or the circumference cannot exist under your logic.
You can stop LARPing as someone with any credibility on the topic now.

[Anonymous No.14799361]

>>14797330
wow yeah im sure adding another dimension will make it easier for the retard to understand

[Anonymous No.14799372]

>>14799354
>You can stop LARPing as someone with any credibility on the topic now.
So far you are the only larper here. The fact is that this way of treating infinite sets of real numbers is just inconsistent and leads to pretty ridiculous conclusions. Meanwhile you offered no arguments apart from "no because I say so".

[Anonymous No.14799385]

>>14791099
We have to first establish what it means for one set to have the same number of elements as another set. This is arbitrary, but what matches most expectations is simply to identify each element of one set with exactly one element of the other set, in such a way that every element of each set is accounted for. We call such an identification a bijection.
We can extend this definition to infinite sets and we come up with some counterintuitive results. For example, there are as many even integers as there are integers; we may prove this with the bijection [math]f:\mathbb{Z}\to\mathbb{Z}[/math] given by [math]k \mapsto 2k[/math]. This is counterintuitive because clearly the set of even integers is a strict subset of the set of integers, but we continue to accept the bijection as a standard for establishing equinumerosity.
Your picture clearly provides a bijection from [math]\mathbb{R}\to\mathbb{R}\times\mathbb{R}[/math]. Incidentally, not all infinite sets have the same cardinality. For example, while one may establish a bijection between the set of naturals and the set of rationals, Cantor's diagonal argument shows that one cannot establish a bijection between the set of naturals and the set of reals; one is "bigger" than the other.
It's okay and perfectly normal if these definitions strike you as odd or meaningless. We often leave out the motivation for definitions, but hopefully I was able to provide a bit of that.

[Anonymous No.14799405]

>>14795376
>Just looked up with a bijection is
this has to be bait

[Anonymous No.14802813]

>>14799385
> [math]f:\Bbb Z \to \Bbb Z\[/math]given by[math]k \mapsto 2k[/math]
Use \text{} next time anon