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.
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
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
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.
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.