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24 sats \ 0 replies \ @noknees 22 Jun \ on: Is Mathematics Mostly Chaos or Mostly Order? science
"To infinity and beyond!" ā Buzz Lightyear
I was reading this article yesterday I really loved Cantor's diagonal argument.
Cantor shattered the illusion that all infinities are equal by introducing uncountable infinity
Suppose you try to list all real numbers between 0 and 1.
Modify each decimal in the nth number's nth place (the diagonal).
This process gives you a new number not in your list, no matter how complete you think it is.
Even if you had eternity, you'd miss some.
āµā (aleph-null) ā countable infinity (e.g. natural numbers).
š (the cardinality of the continuum) ā uncountable infinity (real numbers).
And beyond: Cantor's theorem shows there's always a bigger infinity, by taking the power set.
Pretty simple yet mind bending lmfao