Imagine you have a ruler. You want to cut it exactly at 10 cm mark.
Maybe you were able to cut at 10.000, but if you go more precise you'll start seeing other digits, and they will not be repeating. You just picked a real number.
Also, my intuition for why almost all numbers are irrational: if you break a ruler at any random part, and then measure it, the probability is zero that as you look at the decimal digits they are all zero or have a repeating pattern. They will basically be random digits.
> Maybe you were able to cut at 10.000, but if you go more precise you'll start seeing other digits, and they will not be repeating. You just picked a real number.
A reasonably defensible inference would be that adding a finite amount of precision adds a finite number of additional digits. That is a physically realizable operation. There's no obvious physical meaning to the idea of repeating that operation infinitely many times, so this is not clearly a meaningful way of defining or constructing real numbers. If you were trying to use this construction to convince a skeptic that irrational real numbers exist, you would fail -- they would simply retort that arbitrary finite precision exists and that you have failed to demonstrate infinite non-repeating, non-terminating precision.
Imagine you have a ruler. You want to cut it exactly at 10 cm mark.
Maybe you were able to cut at 10.000, but if you go more precise you'll start seeing other digits, and they will not be repeating. You just picked a real number.
Also, my intuition for why almost all numbers are irrational: if you break a ruler at any random part, and then measure it, the probability is zero that as you look at the decimal digits they are all zero or have a repeating pattern. They will basically be random digits.