quasi
Posts:
9,079
Registered:
7/15/05
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Re: Cardinality of turning wheel
Posted:
Mar 3, 2013 5:04 AM
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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.
quasi
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