quasi
Posts:
12,021
Registered:
7/15/05


Re: Cardinality of turning wheel
Posted:
Mar 3, 2013 5:49 AM


netzweltler wrote: >quasi wrote: >> 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 onetoone correspondence >> >> >> with the set of integers, hence has cardinality aleph0. >> >> >> >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/(t2t1) revolutions per >> unit time. >> >> So yes, the concepts of time and velocity are relevant. > >Is it true to say, that the cardinality of the set of >revolutions of the wheel depends on the velocity? So, can be >aleph_0 or aleph_1?
Assuming distinct revolutions begin at distinct times, the interval of time between consecutive revolutions has positive length, so each such time interval contains a rational time. It follows that the set of revolutions is countably infinite, so has cardinality aleph0.
>See this post: > >On 3 Mrz., 03:17, William Elliot <ma...@panix.com> wrote: >> On Sat, 2 Mar 2013, netzweltler wrote: >> > On 2 Mrz., 20:39, quasi <qu...@null.set> wrote: >> >> > > For a wheel revolving forever (both past and future), the >> > > set of revolutions is in onetoone correspondence with the >> > > set of integers, hence has cardinality aleph0. >> >> > 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? >> >> No. In that case the wheel, necessarily composed of purely virtual >> fantasy, would have spun, in it's eternal life, aleph_1 times.
Of course that's nonsense  Elliot wasn't trying to be serious.
quasi

