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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).
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.
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.
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 :)
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.
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?
Because
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.
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.
Unless they're assuming a certain resolution of measurement.
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
My new years resolution will be to solve this paradox.
