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I feel you.
Don't talk about it, I passed my last math exam of my life (I hope so) in February, I've never thought I'd meet it again. But Volsung's explanation is so good.
:wat:
T-Thank you, I finished the series a month ago and still...
Melitele bless you too.
I've just watched this... epic facepalm and laughing.
[video=youtube;eZM-eG-SZ_A]https://www.youtube.com/watch?v=eZM-eG-SZ_A[/video]
And any of you still asking why I believe I'm a cat instead a fashioned woman?
* 1:07 in memory of @ReptilePZ don't hate me
Embarrassment lv too high, code red. But hey, going to the fashion week is cool, so let's show ignorance as much as we can and pretend we know everything. Exactly.
Cats - Reptiles 1-1
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Some interesting math that's fun to read ;D
https://math.stackexchange.com/questions/733754/visually-stunning-math-concepts-which-are-easy-to-explain
https://math.stackexchange.com/questions/733754/visually-stunning-math-concepts-which-are-easy-to-explain
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I really don't know what to answer about the other people but to me it's ALWAYS cattime-pics!!!!Is it that time again?For cat pics I mean?
Since we have so many cat pics lets throw in some more math. This is meta-mathematics so no calculation involved.
What number is larger than infinity? What size is infinity? We know there is an infinite amount of Natural numbers (0,1,...,infinity). But how many numbers with decimal expansion are there between two natural numbers, such as 0 and 1? 0.1, 0.01, 0.001, etc. Also an infinite amount of numbers. So how can both sets, natural and real, have an infinite amount of numbers when one is clearly larger than the other?
Georg Cantor's approach is very clever. If we have a set such as {1,2,3}, how many combinations of elements from this set can we make? Let's see. {emptyset}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3} and {1,2,3}. That's 8 subsets. If we put them all together { {emptyset}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} } this is called the power set of {1,2,3}, and it has 8 elements as you can see. For larger sets, say sets with n elements, the size (called cardinality) of their power set is 2^n. In the previous example, n = 3 and therefore 2^3 = 8 is the size of its power set. So what happens when we estimate the size of the power set of the Natural numbers? That'd be 2^(infinity). And for any positive value of n, 2^n is necessarily greater than n. So there must be a number that is larger than the size of the set of natural numbers!
This number was named Aleph-null or Aleph-zero by Cantor, and its value is necessarily greater than the "infinity" of the natural numbers as 2 raised to the power of infinity is greater than infinity
This also led to the better understanding of transfinite numbers (larger than infinity).
Since these numbers are just so damn big, substracting 1 shouldn't really matter, right? This leads to very amusing drinking songs like:
"Aleph-null bottles of beer on the wall, Aleph-null bottles of beer, Take one down, and pass it around, Aleph-null bottles of beer on the wall" (repeat).
What number is larger than infinity? What size is infinity? We know there is an infinite amount of Natural numbers (0,1,...,infinity). But how many numbers with decimal expansion are there between two natural numbers, such as 0 and 1? 0.1, 0.01, 0.001, etc. Also an infinite amount of numbers. So how can both sets, natural and real, have an infinite amount of numbers when one is clearly larger than the other?
Georg Cantor's approach is very clever. If we have a set such as {1,2,3}, how many combinations of elements from this set can we make? Let's see. {emptyset}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3} and {1,2,3}. That's 8 subsets. If we put them all together { {emptyset}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} } this is called the power set of {1,2,3}, and it has 8 elements as you can see. For larger sets, say sets with n elements, the size (called cardinality) of their power set is 2^n. In the previous example, n = 3 and therefore 2^3 = 8 is the size of its power set. So what happens when we estimate the size of the power set of the Natural numbers? That'd be 2^(infinity). And for any positive value of n, 2^n is necessarily greater than n. So there must be a number that is larger than the size of the set of natural numbers!
This number was named Aleph-null or Aleph-zero by Cantor, and its value is necessarily greater than the "infinity" of the natural numbers as 2 raised to the power of infinity is greater than infinity
This also led to the better understanding of transfinite numbers (larger than infinity).Since these numbers are just so damn big, substracting 1 shouldn't really matter, right? This leads to very amusing drinking songs like:
"Aleph-null bottles of beer on the wall, Aleph-null bottles of beer, Take one down, and pass it around, Aleph-null bottles of beer on the wall" (repeat).
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Here is a good read https://en.wikipedia.org/wiki/Cantor's_paradox
https://en.wikipedia.org/wiki/Cantor's_paradox
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On a slightly related subject. How can the physical universe (and events in it) be fully described? As a continuum, or as a countable set? For instance, if you assume that all can be enumerated by elementary particles - it can be viewed a countable set. From the point of view of mysics for instance, all the universe and its interactions can be described with "names", i.e. symbolic information which is a countable set too. I.e. imagine an enormous computer which can perform infinite computations of all of those interactions.
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