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OT: Happy Birthday David Hilbert, 159 today.

Hmmm, what if that bus came by, empty this time, and an infinite number of hotel guests check out of the hotel? Will there still be an infinite number of guests in the hotel or will it be empty? I mean, how much is ∞ - ∞ ?
 
I mean, how much is ∞ - ∞ ?
It’s undefined. Here’s why: Suppose A is the set of all positive integers, B is the set of all positive even integers, and C is the set of all positive integers greater than or equal to 3. Each of A, B, C are infinite sets. But A-B (meaning, what’s left when you remove B from A) is the set of all positive odd integers, an infinite set. On the other hand, A-C is equal to the set {1,2}, which has cardinality 2, finite. You can’t get a definite answer, so it’s undefined.
 
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It’s undefined. Here’s why: Suppose A is the set of all positive integers, B is the set of all positive even integers, and C is the set of all positive integers greater than or equal to 3. Each of A, B, C are infinite sets. But A-B (meaning, what’s left when you remove B from A) is the set of all positive odd integers, an infinite set. On the other hand, A-C is equal to the set {1,2}, which has cardinality 2, finite. You can’t get a definite answer, so it’s undefined.
So how does Hilbert reconcile this with his hotel? The Hotel California Corundum: an infinite set can check in anytime they'd like but it's not clear whether they can ever leave?
 
It’s undefined. Here’s why: Suppose A is the set of all positive integers, B is the set of all positive even integers, and C is the set of all positive integers greater than or equal to 3. Each of A, B, C are infinite sets. But A-B (meaning, what’s left when you remove B from A) is the set of all positive odd integers, an infinite set. On the other hand, A-C is equal to the set {1,2}, which has cardinality 2, finite. You can’t get a definite answer, so it’s undefined.


Stop making stuff up.
 
So how does Hilbert reconcile this with his hotel? The Hotel California Corundum: an infinite set can check in anytime they'd like but it's not clear whether they can ever leave?
An infinite number can leave the hotel at any time. It’s just that after an infinite number of guests check out, there’s no certainty as to how many guests remain.

Case 1: Everyone checks out. Zero guests remain.

Case 2: Each guest in an even-numbered room checks out. Infinitely many guests remain, those in the odd-numbered rooms.

Case 3: The guest in Room #645389754 remains, and every other guest checks out. One guest remains.
 
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accb09fe17e5ecc25c7a9740b139f4ce.jpg
 
Hmmm, what if that bus came by, empty this time, and an infinite number of hotel guests check out of the hotel? Will there still be an infinite number of guests in the hotel or will it be empty? I mean, how much is ∞ - ∞ ?

Zero, just as one - one = zero!
 
No.

one - one = 0, two - two = 0, three - three = 0, ..., but ∞ - ∞ is undefined.

See LionJim's explanations why.
So infinity is actually like the rats in NYC (or DC). Until you catch them, they don't exist?
 
I liked Schrodinger's Cat better.

It has always fascinated me.

But what I always wondered was, wouldn’t the cat itself be an observer — one that would sure as hell know whether it was still alive or not — and thus collapse the wave function of that potentially decaying atomic particle that would release the poison, meaning that there would be a definite result inside the box even before the experimenter opened it up to look?
 
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It has always fascinated me.

But what I always wondered was, wouldn’t the cat itself be an observer — one that would sure as hell know whether it was still alive or not — and thus collapse the wave function of that potentially decaying atomic particle that would release the poison, meaning that there would be a definite result inside the box even before the experimenter opened it up to look?
Plus, cats are tough mother ****ers. Give them 5% points just for that, right?
 
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Hilbert was the world’s best and most influential mathematician from roughly 1890-1920. He did too much for me to reasonably summarize, so I’ll just post this cool video on the Hilbert Hotel, in which he explained a concept discovered by Cantor.



Pfft, I learned about that in school. I even got a D-.
 
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