Probabilistically it’s true. Pick a number at random from the uniform distribution of values between zero and one. With nearly total certainty, this value cannot be represented except by enumerating every single one of its infinite digits.
What does it mean to pick a number at random between zero and one?
Does "picking a number at random" even make sense for an infinitely large set where you can't describe most items? Isn't "picking" the act of describing an item? (This might sound like a stupid question but I'm sure there is a mathematical definition of "picking")
If I pick a number at random using some method for picking that requires me to identify what I picked then 100% of the time I'll get a number I can identify, such as the number that is the solution to x^2=2, or the ratio between a square and a circle, or the quotient of 3 and 7. All those numbers I can't describe will never be picked.
I can do infinitely many coin flips and say the number I picked has the decimals described by that binary sequence. But I'd never be done picking...
Do you know measure theory? It gives you a formal definition of "almost all" or "almost surely" based on subsets which have the same measure as the full set they're in. Like the irrational numbers between 0 and 1.
