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:

  • Kogasa
    2•11 months ago

    The best part of this is that it is possible to construct a set that has the same cardinality as the real numbers but is not dense, such as the Cantor set.

    Well that’s not as hard as it sounds, [0,1] isn’t dense in the reals either. It is however dense with respect to itself, in the sense that the closure of [0,1] in the reals is [0,1]. The Cantor set has the special property of being nowhere dense, which is to say that it contains no intervals (taking for granted that it is closed). It’s like a bunch of disjointed, sparse dots that has no length or substance, yet there are uncountably many points.