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Comment on Jeopardy wall calendar pretending that the coastline paradox doesn't exist

BetterDev@programming.dev ⁨7⁩ ⁨hours⁩ ago

Its true that not all intinities 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.

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Atlas_@lemmy.world ⁨5⁩ ⁨hours⁩ ago
I’ll label every real number with the integer 1.

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theunknownmuncher@lemmy.world ⁨7⁩ ⁨hours⁩ ago

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.

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- BetterDev@programming.dev ⁨7⁩ ⁨hours⁩ ago
  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 :)
  
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  - almost1337@lemmy.zip ⁨4⁩ ⁨hours⁩ ago
    And then aleph numbers get thrown into the conversation
    
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  - theunknownmuncher@lemmy.world ⁨6⁩ ⁨hours⁩ ago
    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
    
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    BetterDev@programming.dev ⁨5⁩ ⁨hours⁩ ago
    Yeah, if you use an arbitrary standardized measuring stick, the problem goes away, as it is no longer infinite.
    
    Still a fun throught experiment to demonstrate how unintuitive infinities are!
    
    Anyway, major kudos to you for engaging with this thread in good faith! Its so rare these days, I barely venture to comment anymore. Respect.
    
    … and thank you for the opportunity to share a weird math fact!
    
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  - BetterDev@programming.dev ⁨7⁩ ⁨hours⁩ ago
    And it may very well be true, but we can’t prove it mathematically.
    
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