@ZerkorNotToZerk said in #10:
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@corvusmellori thanks, so I guessed it correctly, at least for the smallest? I am no mathematician, but I like it. What I wrote about the biggest one, do you think this makes sense?
Biggest One doesn't logicly Exists. I know about measurement of Infinity as Cardinals, but logicly the don't put limits on anything
@ZerkorNotToZerk said in #5:
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@WassimBerbar I know that one, maybe I need a refresher.
Imagine a huge hotel with an infinite number of rooms. All the rooms are full.
One dude wants a room. Every room is full, so the manager asks everyone to move up one room (dude 1 goes to room 2, ddude 2 goes room 3 etc.) then the 1st room is empty so the new guest goes into that room. Proving infinity+1 = infinity.
An infinite number of dudes want a room. Every room is full, so the manager asks everyone to move to the room double his original room (dude 2 goes to room 4, dude 3 goes to room 6 etc.), so the people will take the rooms with the odd number. Proving infinity + infinity = infinity.
I don't remember the rest of the story.
Basically: take the singularity of a blackhole, it is infinitely small, because you cant reach the singularity, not because it is infinitely small (even tho it can be, not sure), many scientists believe is it infinitely deep. So, we dont know what the smallest infinity is, (because there are two theories, infinitely deep and infinitely small).
Tho the largest infinite, I am also not sure, the universe might be infinitely large, because we dont know what's past the ''Observable Universe'', infinitely large or does it not stop?
2 is the smallest number. 0 and 1 don't count. Consider;
"I have a number of cats."
"Oh, how many?"
"0"
"0 cats isnt a number of cats silly"
"Ok fine, I have 1 cat."
"1 cat is not a number of cats! ARGH"
"I have 2 cats."
"That makes sense. 2 is a number of cats"
"Actually I have 4 halves of cats."
Then, an infinite number of buses turn up, each bus with an infinite number of dudes, all of whom want a room. Every room is full, so the manager seeks some way of organising all the people who have arrived, so that he can guarantee every single person will eventually be counted. Then, he can pull the same trick he did earlier (move to the room that's double your room and fill up the empty spaces).
Let (x,y) denote the y'th passenger on board the x'th bus (x,y positive integers). He does a "zig-zag" pattern to organise all the passengers into a list, so that every passenger is counted: first is (1,1), then (1,2), then (2,1), then (3,1), then (2,2), then (1,3), then (1,4)... (this is best understood with a diagram)
Once he has this list, which contains every single passenger, he assigns them to the odd-number rooms. This proves that infinity*infinity + infinity = infinity, where "infinity" represents a countable infinity (aleph-null).
Lastly, an infinite number of dudes turn up, but this time they are all labelled with numbers between 0 and 1, where no two people have the same number. It turns out to be impossible for the manager to fit all the people, even if the hotel is initially empty. This is because it's impossible to have an organised list with every single passenger on said list, which is necessary for the manager to begin to assign rooms.
Assume for a contradiction that such a list does exist, with every single passenger's number on it. But then, take the first digit of the first number on the list, the second digit of the second number, and so on... and add one to each of these digits (if it's a 9, make it a 0) and then put it all together to form a new decimal. This clearly cannot be on our list, since at least one of its digits differs from every single number on our infinite list. Therefore our list cannot ever contain every single passenger's number.
For example, if our list was:
0.13948729347... (first digit 1)
0.2035718952... (second digit 0)
0.92183429137... (third digit 1)
0.475610023904... (fourth digit 6)
0.129569135791.... (fifth digit 6)
and so on...
our new decimal would begin 0.21277.... which cannot be equal to any of the numbers in our infinite list.
This proves that there exists an infinity greater than the countable infinity, called the uncountable infinity (aleph-one)
We are not talking about the smallest number tho..that is a different question.
@MercuryTrismegistus<Comment deleted by user>
@four_legs_good can you give me a feedback, if the way I defined "omega" was it right?
<Comment deleted by user>
basically: bit integer limit positive and bit integer limit negative
