Exploring an Infinitely Accurate Watch: Time Without End?

[F-Bert]

when i was a kid, i stared at a digital watch, and imaigned that the numbers did not stop at the seconds, but went on , so that you had milli-seconds, billi-seconds trilli-seconds(probably not the right terms for them) and so on. i imaigne you could continue this series forever, just get smaller units of time each progressive column of numbers
This extremely accurate watch had an obvious rule to me: when one column of numbers reached its max( 60 seconds, 60 minutes..etc) the column to its left would increase by one. so when you had 60 seconds, you had one more minute. but if there was an infinite set of rows, wouldn't time cease to run, because each column would be "waiting" for the column to its right, which would be waiting for the column to its right and so on?
i think that this would probably not be a flaw of time, but of the time-keeping method.
any thoughts that could help?


P.S. I am not sure if i posted this in the right section

 

[kokain]

That sounds like zeno's paradox. Walking from point A to point B would require you to reach a point half of the distance. To reach a point half of this distance you would first need to reach a point half of the distance to that point and half of that distance and half of that distance andf half of that distance and half of... you get the idea. You would not be able to move. Thinking in those terms is definatly fun.

 

[hypnagogue]

This is indeed a variation on Zeno's paradox, and it can be treated in the same way by using the concept of limits to remove the apparent paradox.

The value at any particular column i in the time keeper must wait for the value in the next column i + 1 to reach a value of, say, 10, before it incerements. Likewise, column i + 1 must wait for column i + 2 to reach a value of 10 before it increments. So if we assume an infinite number of columns, it looks as if we have a sort of infinite sum of time units on our hands, and therefore that the value of any column j never increments because it must wait on the passage of an infinite amount of time.

The fault in this reasoning is that it assumes infinite sums must produce infinite (indefinitely high) values. But any familiarity with calculus will show that we can have infinite sums that nevertheless converge to a particular limiting value. It is possible for an infinite sum to converge to a definite value if the successive terms in the sum decrease sufficiently fast. For instance, the infinite sum 3/10 + 3/10² + 3/10³ + ... converges to the value 0.333... = 1/3.

With this in mind, let's look at the seconds column in our time keeper, supposing that there are an infinite number of decimal columns following the seconds column, and that all these digits are initially set to 0. Will the value of the seconds column ever increment to 1? Yes, it will, because the values of the time units in the following columns decrease sufficiently fast to allow a converging sum. In particular, the infinite sum 9/10 sec + 9/100 sec + 9 /1000 sec + ... = 0.999... sec = 1 sec, and thus we have our incrementation of the seconds column in a finite amount of time. (For various discussions on why 0.999... = 1, see this thread.)