Date: Mar 5, 2013 4:19 PM
Subject: Re: Cardinality of turning wheel
>> 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 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 xy-plane, lying on the
x-axis and rolling, to the right say, along the x-axis.
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
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 anti-intuitive as to be rejected as nonsense.
Any such proposed model can be an idealized model, not
necessarily matching what happens in the real-world. 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 real-world, 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 anti-intuitive 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.