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Topic: Cardinality of turning wheel
Replies: 43   Last Post: Mar 10, 2013 1:55 AM

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 netzweltler Posts: 425 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 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.
>
> >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 aleph-0.
>
>
>
>
>

> >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 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?

>
> >> 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?

Date Subject Author
3/2/13 netzweltler
3/2/13 Frederick Williams
3/2/13 quasi
3/2/13 netzweltler
3/2/13 William Elliot
3/3/13 quasi
3/3/13 netzweltler
3/3/13 quasi
3/3/13 netzweltler
3/3/13 quasi
3/3/13 netzweltler
3/3/13 Brian Chandler
3/4/13 netzweltler
3/3/13 quasi
3/3/13 Frederick Williams
3/3/13 quasi
3/4/13 netzweltler
3/4/13 quasi
3/4/13 Shmuel (Seymour J.) Metz
3/5/13 Frederick Williams
3/5/13 netzweltler
3/5/13 quasi
3/6/13 netzweltler
3/6/13 quasi
3/7/13 netzweltler
3/7/13 quasi
3/8/13 netzweltler
3/8/13 quasi
3/8/13 netzweltler
3/8/13 quasi
3/8/13 Frederick Williams
3/2/13 Frederick Williams
3/3/13 Frederick Williams
3/5/13 K_h
3/7/13 Frederick Williams
3/7/13 Frederick Williams
3/3/13 Shmuel (Seymour J.) Metz
3/7/13 Frederick Williams
3/10/13 Shmuel (Seymour J.) Metz