quasi
Posts:
9,079
Registered:
7/15/05
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Re: Cardinality of turning wheel
Posted:
Mar 3, 2013 5:49 AM
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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
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