What concepts or facts do you know from math that is mind blowing, awesome, or simply fascinating?

Here are some I would like to share:

  • Gödel’s incompleteness theorems: There are some problems in math so difficult that it can never be solved no matter how much time you put into it.
  • Halting problem: It is impossible to write a program that can figure out whether or not any input program loops forever or finishes running. (Undecidablity)

The Busy Beaver function

Now this is the mind blowing one. What is the largest non-infinite number you know? Graham’s Number? TREE(3)? TREE(TREE(3))? This one will beat it easily.

  • The Busy Beaver function produces the fastest growing number that is theoretically possible. These numbers are so large we don’t even know if you can compute the function to get the value even with an infinitely powerful PC.
  • In fact, just the mere act of being able to compute the value would mean solving the hardest problems in mathematics.
  • Σ(1) = 1
  • Σ(4) = 13
  • Σ(6) > 101010101010101010101010101010 (10s are stacked on each other)
  • Σ(17) > Graham’s Number
  • Σ(27) If you can compute this function the Goldbach conjecture is false.
  • Σ(744) If you can compute this function the Riemann hypothesis is false.

Sources:

  • wafergirl
    1·
    1 year ago

    Yes, all of this is super cool! My favorite part, though, is the continuum hypothesis! A question that went unsolved for a long time is whether or not there exists a set with cardinality in between that of the natural numbers and the real numbers. This problem was eventually solved, and the answer is extremely interesting! The answer is not that the hypothesis is true or false, but that the hypothesis is independent from our currently accepted axioms of math. So, we could have a perfectly valid version of math where the hypothesis is true, as well as a perfectly valid version where it’s false. This opens up a whole new world of possibilities for set theory, and there is still active research being done on the consequences that different axioms have on the relative sizes of uncountable cardinals that are less than or equal to the size of the reals.