quasi
Posts:
10,660
Registered:
7/15/05


Re: Cardinality of turning wheel
Posted:
Mar 5, 2013 4:19 PM


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 it true to say, that the wheel finishes countably >infinitely many revolutions, whenever we assign an origin (the >point between past and future)?
It all depends on starting assumptions (axioms).
Assume the wheel is a circle in the xyplane, lying on the xaxis and rolling, to the right say, along the xaxis.
Assume an instant of time corresponds to a point on a real number line (the time axis), and a unit time interval is any interval of the form [t,t+1] on the time axis.
Define the speed of the wheel at a given time t, as the speed at time t of the center of the wheel, regarded as a point particle.
Assume that for all times t, the speed v(t) of the wheel is positive and bounded within fixed positive limits, that is, assume that there exist positive real numbers a,b such that, for all t, a <= v(t) <= b.
With those assumptions, it's immediate that the set of revolutions is countably infinite.
If you change the axioms of the model, you can get different results. For the original question, the assumptions were not fully specified, so I chose what I thought were reasonable assumptions in an idealized model. With those assumptions, the set of revolutions is countably infinite.
The OP then asked if the set of revolutions would still be countably infinite if the speed of the wheel was infinite. As I then indicated, I felt that it would be hard to specify such a model, with a full set of assumptions making clear what is meant by time, space, velocity, a wheel, a revolution of the wheel, in such a way as to not to be so antiintuitive as to be rejected as nonsense.
Any such proposed model can be an idealized model, not necessarily matching what happens in the realworld. On the other hand, terms like time, space, velocity, time, wheel, revolution have intuitive meanings (based on how we use those terms in the real world), and hence any model which redefines those terms in ways that are not even close to common intuition needs to be justified.
For example, maybe the claim is that, contrary to intuition, the new model actually _does_ match what happens in the real world. Such a claim, if some evidence is provided, might justify the analysis and testing of the model.
Even if the model is not realworld, the justification may simply be that the model is mathematically interesting. Fine, but still the choice of meanings for time, space, etc. shouldn't be so antiintuitive as to be hopelessly confusing.
As far as the notion of infinite speed, I see the specification of such model as problematic, but as I said, I would be willing to look at a proposal for such a model, so long as the assumptions were fully specified, and sufficient justification for analyzing the model was provided.
quasi

