It’s almost certainly a rational number as activity in the body is finite. There will be an exact point at which a certain molecule has interacted with another molecule marking the exact point of perfect sleep. Averaging many rational numbers can’t get you an irrational number.
Also it is invalid to compare infinite series like you do in your paracentesis argument, there are infinitely many irrational numbers and infinitely many rational ones. If you do it, you run in to contradictions. (Something which is related to Cantor's paradox)
The statement OP put in parentheses is a well-defined statement in mathematics. It means that the Lebesgue measure of the set of the rational numbers is zero in the space of real numbers.
"If the physical property that time meassuring devices meassure is continuous, it must also contain irrational numbers?" Is that what you are refering to?
Mabye... To me it seams like nothing is truly continuous i nature. But mabye there is such a thing somewere out there somewere.
But irrational numbers require definitions that contain or require recursion. Mabye physical time is built with such a recursive definition?
"There will be an exact point at which a certain molecule has interacted with another molecule" statement is contradictory with the uncertainty associated with time as described by quantum mechanics and molecular interactions due to Brownian motion. Material that is returned when searching for those topics will better answer your further questions about them and those above than I here.
The uncertainty principle is about unavoidable measurement "errors" (measurements that can’t be done). If you are going to measure the time (or anything) you will always get finite values. Finite values are not irrational.
Either one talks about the underlying physics or about the measurements of it. If one talks about the underlying physics all bets are of, it is unmeasurable by definition. Anything is possible bellow the measurement threshold(including irrational numbers).
If one talks about the measurements you will always get finite rational values.
Also it is invalid to compare infinite series like you do in your paracentesis argument, there are infinitely many irrational numbers and infinitely many rational ones. If you do it, you run in to contradictions. (Something which is related to Cantor's paradox)