netzweltler
Posts:
473
From:
Germany
Registered:
8/6/10


Re: Cardinality of turning wheel
Posted:
Mar 3, 2013 5:57 AM


On 3 Mrz., 11:49, quasi <qu...@null.set> wrote: > 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
Meaning, that the notion of infinite speed is nonsense, right?

