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.
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...