Re: An uncountable countable set

In article <ec9ssu$9km$1@xxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

Virgil wrote:
In article <ec9nrs$44k$2@xxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

Virgil wrote:
In article <ec8h1q$prm$1@xxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

It would seem to me that any good theory
of infinite sets could be applied to infinite sets of points, such as
the reals in (0,1] or those in (0,2], and be able to draw conclusions
such as that there is twice as many points and twice as much space in
the second as in the first
That would require each point to occupy enough space that only a finite
number of them could fit into any finite interval.
WRONG!

It requires that some relationship be set up between a particular
infinity of points and a finite length. What you suggest would be a
finite set of finite elements, not any kind of infinite set.

If TO insists that infinitely many points can be compressed into a
finite space, then one can just as easily compress twice as many points
into the same space.

If one takes the points of (0,2] and places each point from x at
position x/2 in (0,1], one has compressed "twice as many" points into
(0,1] as were originally there. Thus there cannot be any fixed
proportionality between the "number of points" in an interval and its
length for intervals of positive length.

There is when one declares it.

The problem being that one can declare as many different ones as one
chooses to declare, and each is just as valid as any other.

Big'un is the number of reals in the unit
interval, and the number of unit intervals on the infinite real line.

I declare Card(P(N)) to be the number of reals in the unit interval and
Card(N) to be the number of unit intervals with disjoint interiors in
the infinite real line.

TO's definition is either wrong (if his unit intervals are to have
disjoint interiors) or he has corrupted his own theory by letting the
umber of points in the unit interval equal the number of points in the
entire real line (if overlapping intervals are allowed, the number of
unit intervals in the real line equals the umber of points in the real
line).


Where you claim to compress or stretch the inherent density of the real
points on the line, you make proper measure impossible.

Is TO going to pontificate on measure theory on the real line?

If so how is he going to deal with the inevitable non-measurable sets?

Cardinality has no problem with them.
.

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