netzweltler wrote: >quasi wrote: >> netzweltler wrote: >> >quasi wrote: >> >> netzweltler wrote: >> >> >What is the cardinality of the number of revolutions of a >> >> >turning wheel, if there is no beginning and no end to it? >> >> >> For a wheel revolving forever (both past and future), the >> >> set of revolutions is in one-to-one correspondence with the >> >> set of integers, hence has cardinality aleph-0. >> >> >Is this still true, if the wheel is revolving at infinite >> >speed, meaning, that we can see at least one revolution no >> >matter how small the time we are watching it? >> >> That's totally inconsistent with my intuition about velocity >> and time. > >Do we need to define 'velocity' and 'time'? Do we need to >assign an origin, past and future to give a valid answer to the >question "What is the cardinality of the number of revolutions >of a turning wheel, if there is no beginning and no end to it?"
As I see it, revolutions correspond to time points on the number line.
The concept of perpetual revolution without beginning or end implies that for each revolution, there is a previous one and a next one. Hence if two consecutive revolutions occur at times t1 and t2 with t1 < t2, the average rotational velocity for the time interval [t1,t2] is 1/(t2-t1) revolutions per unit time.
So yes, the concepts of time and velocity are relevant.
>> I assumed that instants of time could be represented >> one-to-one as points on the real number line, and that there >> exist real numbers a,b with 0 < a <= b such that at any >> instant of time, the velocity v of the wheel, expressed in >> revolutions per unit time, is between a and b inclusive.