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.
this post was submitted on 20 Nov 2025
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But how do we know Asia's coastline isn't more jaggedy?
Because
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.
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.
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
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.
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.
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
Is it?
The main problem with a coastline's shape isn't the fractality of it, or the relative size we measure in (the "resolution").
It's the fact that a coastline isn't a static thing. The tides move the shoreline by up to a few meters.
Then there are tectonic movements. These are much slower, but much more powerful: at one point Asia wasn't even a thing.
As you take the "resolution" up, yes - you'll see various fractal-like behaviour.
But, and thus is a big but: this will happen even if you take a straight ruler of, say, 1m in length (or, since we have to deal with every little edge case here, the part of it that actually measures out a meter). If you zoom in on it at the molecular and atomic levels, you'll come across the same problem: a straight line isn't a straight line! Just by taking an optical microscope, you'll see the inherent jaggedness (fractality) of our supposedly-straight ruler. It turns out our ruler just appears straight at the human "resolution" (scale).
But does that mean a ruler measuring out 1m isn't 1m long? While it may not have tectonic or tidal movements, the molecules building up the ruler aren't straight.
Does this mean the ruler, "zoomed in enough", will appear to be of infinite length?
Yes.
But does that mean its length is infinite?
No. Its length is clearly 1m, +/- a small rounding error.
The same idea applies to our coastline.
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.
What is the difference between length and area, other than the one dimension they are apart?
What you're taking as a common sense assumption for area is equally applicable to the length. Find two extremes, and the answer is somewhere in the middle. The less extreme those extremes become, the more accurate the approximation.
Just as you can integrate the area, there must be an equivalent process to integrate the length.
And besides, any curve used to model the length of a coastline is a bigger assumption than a sufficiently sane "resolution" used to divide the curve into discrete intervals for the purposes of geodesic measurements. As you vary the number of reference points,the length will indeed increase. But after each successive round of refinement, the difference will be less and less, even though it will consistently rise. At one point, it will become insignificant enough.
Why does area get to be especially fun and definite while length, its one-dimension-away sibling doesn't?
What about volume? Is it an unsolveable enigma like length, or a long-solved problem like area?
Why does area get to be especially fun and definite while length, its one-dimension-away sibling doesn't?
Excellent question, and as you yourself allude to, it's a question of bounds. If you can establish and upper and lower bound on a quantity and make them approach eachother, you can measure it.
On a finite 2d surface you can make absolute lower and upper bounds on any area - lower is zero, upper is the full surface. All areas are measurable. But on the same surface you can make a line infinitely squiggly and detailed, essentially drawing a fractal. So the upper bound on the length of a line is infinite. Which means not all lines have a measurable length. And that comparing two line lengths might become the same problem as comparing to infinities of the same type, which is not well defined.
This extends naturally to higher dimensions - in a finite 3d space, volumes must be finite, but both lines and areas can be fractally complex and infinite. And so on.
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?
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.
They are literally both ℵ~1~ though?
Aren’t the number of real numbers and the number of integers also infinite? But they aren’t considered equal. The infinite for real numbers is considered larger.
Yes, the number of Intergers is ℵ~0~, the number of real numbers ℵ~1~, and this is what people generally mean with some infinities are bigger than others. Infinities can also be seem bigger than another, but be mathematically equal. The number of natural, real and rational numbers are all infinite, and might seem different, but they are all proven ℵ~0~.
Claypidgin was talking about the real numbers between [0,1] and [0,2], which are both ℵ~1~ infinite. Some infinities are indeed bigger than others, but those 2 are still the same infinity.
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.
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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.
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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.
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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.
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