Re: infinity

David R Tribble said:
> Tony Orlow (aeo6) wrote:
> >> What consistency! What was the contradiction in the notion that [0,1) is
> >> half the size of [0,2) again?
> >
> David R Tribble said:
> >> If you're right, you should be able to name twice as many points
> >> from [0,2) than I can name from [0,1).
> >>
> >> But while you can name two points in [0,2) for every point I choose
> >> in [0,1), I can also name two points in [0,1) for every point you
> >> choose in [0,2).
> >>
> >> b = a x 2, for all a in [0,1)
> >> and b in [0,2)
> >>
> >> c = d / 2, for all d in [0,2)
> >> and c in [0,1)
> >
> Tony Orlow (aeo6) wrote:
> > Your mapping function demonstrates the ratio of 2 between these two sets of
> > reals.
>
> Well, it demonstrates the ratio between the numeric values of
> corresponding members of the two sets.
>
> It also demonstrates that the two sets have exactly the same number
> of members; any given member of one set will have a numeric value
> that is exactly twice or half of the corresponding member in the
> other set.
That's not at all the way I interpret it. If the two sets have the same range
of values, then the one with the lower density or frequency will have fewer
members. When you draw a bijection between the two sets using a mapping
function that doubles one to get the other, you are also doubling the value
range of the first to get the value range of the second, so you get the same
number of elements in twice the space. If the two sets cover the same range,
then they will not have the same number of elements.
>
> Unless, of course, you can provide members of one set that don't
> correspond to any members of the other set. Find a b that has
> no corresponding a, or a d that has no c.
if the two sets have the same range, then the second half of the first set will
have no corresponding members in the second set.
>
>

--
Smiles,

Tony
.

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