Leaving aside Planck scale, infinities can be larger than other infinities.
https://www.cantorsparadise.com/why-some-infinities-are-larger-than-others-fc26863b872f
So when that kid said “well I hate you infinity plus a million” he was on to something mathematically?
No, but if he said “well I hate you two to the power of infinity” he would be.
If your unit of measurement is 1 Asia coastline, all others would be some changing fraction thereof. Mathematical equation paradox maybe but hardly over that disproves the answer.
But how do we know Asia’s coastline isn’t more jaggedy?
Surely the coast of a continent of a given area can only have a finite theoretically maximum length even if the whole coast is a Hilbert Curve filling that area, because the minimum feature size is determined by the surface tension of water m
Hmm, just because the distance measured varies based on the increments it is measured in doesn’t mean that using the same stick it wouldn’t be bigger.
Nah, that’s silly. Asia obviously has the longest coastline.
Sure, based on that paradox, the specific measurement of a given coastline will differ. But if you pick a standard (i.e., 1km straight lines), Asia is easily the longest. Doesn’t matter what standard you pick.
The only way the paradox matter here is of you pick different standards for different coastlines. Which, os obviously wrong.
Some infinites are larger than other infinites.
It’s not a true fractal, so the length has some finite bounding. It’s just stupidly large, since you are tracing the atomic structure.
Let F be a geometric object and let C be the set of counterexamples.
F is a True Fractal ⟺ F satisfies all properties P₁, P₂, …, Pₙ
Where for each counterexample c ∈ C that satisfies P₁…Pₙ: Define Pₙ₊₁ := “is not like c”
The definition recurses infinitely as new counterexamples emerge.
Corollary: Coastlines exhibit fractal properties at every scale… except they don’t, because [insert new property], except that’s also not quite right because [insert newer property], except actually [insert even newer property]…
□ (no true scotsman continues fractally)
This motherfucker coming correct with subscripts.
That’s a fair point. I forgot that some infinites are larger than other infinites.
Did you also forget about Dre?
Did you forget about the game?
Isn’t it a bit like saying “there’s obviously more real numbers between 0 and 2 than between 0 and 1”? Which, to my knowledge, is a false statement.
It isn’t.
When you look at the number of real numbers, you can always find new ones in both - you’ll never run out.
That being said, imagine (or actually draw) two number lines in the same scale. One [0,1] the other [0,2]. Choose a natural number n, and divide both lines with that many lines. You’ll get n+1 segmets in both lines.
When you let n run off into infinity, the number of segments will be the same in both lines. This is the cardinality of the set.
But for practical purposes of measuring a coastline, this approach is flawed.
Yes, you’ll always see n+1 segments, but we aren’t measuring the number of distinct points on the coastline, but rather its length, i.e. the distance between these points.
If you go back to your two to-scale number lines and divide them into n segments, the segments on one are exactly two times larger than on the other.
This is what we want to measure when we want to measure a coastline. The total length drawn when connecting these n points (and not their number!) as the number of points runs off towards infinity.
The solution to this “paradox” is probably closer to the definition of the integral (used to measure areas “under” math functions) than to that of the cardinality of infinite sets (used to measure the number of distinct elements in a set).
But isn’t the issue that coastlines have a fractal nature? That depending on your resolution, you could have a finite or infinite length of a coastline? In which case measurement is hard to define.
Talking about integrals, the fun part is that even with a coastline of indeterminate length, the area of a continent is easy to define to arbitrary precision - you can just define an integral that’s definitely inside the area and one that’s definitely outside the area, and the answer is between those two.
If between 0 and 1 are an infinite number of real numbers, then between 0 and 2 are twice infinite real numbers, IIRC my college math. I probably don’t.
In math they’d both be equal
Discrete math typically teaches that some infinities are greater than others.
Yes, but those are both the same infinite according to math, so no, they’re still equal.
? But they’re not the same infinity according to math.
Infinities don’t care about the actual numbers in the set, but about the cardinality (size). Obviously the numbers between 0,1 and 1,2 are different but have the same size.
But 0,1 and 0,2? Size is unintuitive for infinities because they are … infinite. So the trick is to look for the simplest mathematical formula that can produce a matching from every number of one set to every number in the second. And as somebody has said, every number in 0,2 can be achieves by multiplying a number in 0,1 by 2. So there is a 1 to 1 relation between 0,1 and 0,2. Ergo they are the same size.
Here’s the proof: for each number between 0 and 1, double it and you get a unique number between 0 and 2. And you can do the reverse by halving. So every number in the first set is matched with every number in the second set, meaning they’re the same size.
They are literally both ℵ1 though?
The cardinality of the two intervals [0,1] and [0,2] are equivalent. E.g. for every number in the former you could map it to a unique number in the latter and vice versa. (Multiply or divide by two)
However in statistics, if you have a continuous variable with a uniform distribution on the interval [0, 2] and you want to know what the chances are of that value being between [0,1] then you do what you normally would for a discrete set and divide 1 by 2 because there are twice as many elements in the total than there are in half the range.
In other words, for weird theoretical math the amount of numbers in the reals is equivalent to the amount of any elements in a subset of the reals, but other than those weird cases, you should treat it as though they are different sizes.
Sure, the length of the intervals is easily compared. But saying
there are twice as many elements in the total than there are in half the range
is false. They are both aleph 1. In other words, for each unique element you can pick from [0,2], I can pick a unique element from [0,1]. I could even pick two or more. So you can’t compare the number of elements in the two in a meaningful way other than saying they both belong to the same category of infinite.
This is the whole crux of the coastline problem, isn’t it?
Funny that so many uses of maths depends on measurement, and yet so many pure mathematicians seem to be clueless about how we actually measure things and why its useful. It doesn’t even matters about all this bullshit about infinities , were talking about the real world. It’s all about the precision of the tape measure. Here’s a true story from back in the day:
English Mathematician: You’ll need an infinite number of bricks to build a wall around any island’s coastline. French guy: come on over and see Mont Saint Michel it’s vraiment genial!
English Mathematician: Oh that wall is infinitely far away from the true coastline, those bricks are not regulation infinitesimal length. If they’d started from the other corner they’d have got a different shape, and for sure needed infinite number of infinitesimal bricks to actually build that wall. Sloppy french masons. I can prove it I’ll blast them all away with cannon fire until the glorious mathematical truth is revealed underneath.
One year later French inhabitants: fuck off english maths whore!
Ten years more laterer Hi french dudes! I’m back with a greater number of even bigger state of the art truth seeking cannon. I will prove this if its the last thing i do.
One year later . . .
i hate the coastline ‘paradox’ and every other ‘paradox’ that’s just a missing variable. “if we measure with a big resolution it’s a smaller number of units and a small resolution is a bigger number!?” that’s not a paradox. that’s just how that variable works always. it’s not confusing or interesting at all.
But if you shrink the “yardstick” down to an infinitesimally small size, the length, effectively, becomes infinite… and it’s the same for all coastlines. They’re all infinitely long.
… but some are longer than others. ;)
Literally no. Very hard to measure, but strictly still a finite length. Limits and all that jazz.
Limits can resolve to infinity. The coastline paradox is just the observation that the (semi-reasonable) assumption that landmasses are fractal shaped implies the coastline tends towards infinity with smaller yardsticks.
They can… I wasn’t saying they couldn’t… I meant that as to point to the logic you’d use to prove it finite
My bad for the poor wording though.
Max Planck says no…
Didn’t calculus solve this stuff?
Surely the distance approaches some finite value.
You can’t shrink the yardstick down to an infinitesimal size.
Coastlines are not well defined. They change in time with tides and waves. And even if you take a picture and try to measure that, you still have to decide at what point exactly the sea ends and the land starts.
If the criteria for that is “the line is where it would make a fractal” then sure, by that arbitrary decision, it is infinite. However, a way better way to answer the question “where is the line” is to just decide on a fixed resolution (or variable if you want to get fancy), which makes the distinction between sea and land clearer.
It is like saying that an electron is everywhere in the universe, because of Heisenberg’s uncertainty principle. While it is very technically true, just pick a resolution of 1mm^3 and you know exactly where the electron is.
Not all infinities are equal, friend. Asia does have more infinite coastline than other continents.
Its true that not all infinities are equal, but the way we determine which infinities are larger is as follows
Say you have two infinite sets: A and B A is the set of integers B is the set of positive integers
Now, based on your argument, Asia has the largest infinite coastline in the same way A contains more numbers than B, right?
Well that’s not how infinity works. |B| = |A| surprisingly.
The test you can use to see if one infinity is bigger than another is thus:
Can you take each element of A, and assign a unique member of B to it? If so, they’re the same order of infinity.
As an example where you can’t do this, and therefore the infinite sets are truely of different sizes, is the real numbers vs the integers. Go ahead, try to label every real number with an integer, I’ll wait.
I’ll label every real number with the integer 1.
Go ahead, try to label every real number with an integer, I’ll wait.
Why would I be trying to do this though? You’ve got the argument backwards.
Is the set of all real numbers between 0 and 100 bigger or equal to the set of all real numbers between 0 and 1?
It seems like I’m wrong though and these sets are the same “size” lol
Exactly! It is unintuitive, but there are as many infinite elements of the set of all real numbers between 0 and 1, as there are in the set between 0 and 100.
I hope this demonstrates what the people here arguing for the paradox are saying, to the people who are arguing that one is obviously longer.
Just because something is obvious, doesn’t make it true :)
And then aleph numbers get thrown into the conversation
Alright, I concede. I did it wrong but still ended up with the right answer. There are other responses in this thread with correct explanation for why Asia has more coastline
Yeah, if you use an arbitrary standardized measuring stick, the problem goes away, as it is no longer infinite.
Still a fun thought experiment to demonstrate how unintuitive infinities are!
Anyway, major kudos to you for engaging with this thread in good faith! That is so rare these days, I barely venture to comment anymore. Respect.
… and thank you for the opportunity to share a weird math fact!
Hmm I’ve consulted a mathematician that I know, and they say that cardinality isn’t really the same as “size”, but comparing the two infinite sets of the same cardinality is basically meaningless because infinity is not a “number”, even though one set is provably “bigger” than the other set
And it may very well be true, but we can’t prove it mathematically.
It’s correct, though. You’d apply the same scale of measurements to all coastlines, and using a standard of 1km or 0.5km plot points, Asia wins.
Unless they’re assuming a certain resolution of measurement.
My new years resolution will be to solve this paradox.
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