this post was submitted on 23 Oct 2024
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Combinatorics scares me, the immense size of seemingly trivial things.
For example: If you take a simple 52 card poker deck, shuffle it well, some combination of 4-5 riffles and 4-5 cuts, it is basically 100% certain that the order of all the cards has never been seen before and will never been seen again unless you intentionally order them like that.
52 factorial is an unimaginable number, the amount of unique combinations is so immense it really freaks me out. And all from a simple deck of playing cards.
Chess is another example. Assuming you aren't deliberately trying to copy a specific game, and assuming the game goes longer than around a dozen moves, you will never play the same game ever again, and nobody else for the rest of our civilization ever will either. The amount of possible unique chess games with 40 moves is far far larger than the number of stars in the entire observable universe.
You could play 100 complete chess games with around 40 moves every single second for the rest of your life and you would never replay a game and no other people on earth would ever replay any of your games, they all would be unique.
One last freaky one: There are different sizes of infinity, like literally, there are entire categories of infinities that are larger than other ones.
I won't get into the math here, you can find lots of great vids online explaining it. But here is the freaky fact: There are infinitely more numbers between 1 and 2 than the entire infinite set of natural numbers 1, 2, 3...
In fact, there are infinitely more numbers between any fraction of natural numbers, than the entire infinite natural numbers, no matter how small you make the fraction...
Natural numbers being infinite, how it be possible for the values between 1 and 2 to be "more infinite" ?
It's called countable and uncountable infinity. the idea here is that there are uncountably many numbers between 1 and 2, while there are only countably infinite natural numbers. it actually makes sense when you think about it. let's assume for a moment that the numbers between 1 and 2 are the same "size" of infinity as the natural numbers. If that were true, you'd be able to map every number between 1 and 2 to a natural number. but here's the thing, say you map some number "a" to 22 and another number "b" to 23. Now take the average of these two numbers, (a + b)/2 = c the number "c" is still between 1 and 2, but it hasnโt been mapped to any natural number. this means that there are more numbers between 1 and 2 than there are natural numbers proving that the infinity of real numbers is a different, larger kind of infinity than the infinity of the natural numbers
Your explanation is wrong. There is no reason to believe that "c" has no mapping.
Give me an example of a mapping system for the numbers between 1 and 2 where if you take the average of any 2 sequentially mapped numbers, the number in-between is also mapped.
Yeah, OP seems to be assuming a continuous mapping. It still works if you don't, but the standard way to prove it is the more abstract "diagonal argument".
because I assumed continuous mapping the number c is between a and b it means if it has to be mapped to a natural number the natural number has to be between 22 and 23 but there is no natural number between 22 and 23 , it means c is not mapped to anything