Re: Galileo's Paradox and the Project of the Reals

Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx wrote:
Tony Orlow wrote:

Okay, define me an infinite set which doesn't use successor or order in
the definition, or in the definition of something used in the definition.


The set of all lines in the Euclidean plane.

Define "line" without '<'.


That is an odd but easily satisfied request. "Line" is a primitive in
Euclidean geometry; as such it has no "definition" at all, with or
without '<'.

See:

http://en.wikipedia.org/wiki/Hilbert%27s_axioms#The_Axioms


Or:

The set of all triangles in the Euclidean plane.

Define "triangle" without "line".


Obviously, that would be silly. Presumably, you didn't realize that I
don't "need" '<' to "define" "line" in Euclidean geometry.


Neither of these sets has a "standard" ordering which allows us to say,
for any two elements a, b (lines or triangles) that exactly one of a <
b, a > b or a = b holds true.

Cheers - Chas


No, in the 2D plane, one needs to use something like a lexicographic
ordering by ordering the dimensions of the space, and then using the
order within each dimension.

Sure, one /can/ do something of the sort. But why does one "need" to
order these things at all?

Anyway:

(i) You are talking about Analytic Geometry, in the Cartesian Plane.
The sets I defined are sets of elements from Euclidean Geometry, in the
Euclidean Plane. These are /different things/.

Points in the Euclidean plane are /not/ defined by some pair of real
numbers (x,y). They are primitives, just as lines are. Again, refer to:

http://en.wikipedia.org/wiki/Hilbert%27s_axioms#The_Axioms

where you will fail to find any reference to real numbers, or even
rational numbers.

(ii) It is not in dispute that in general a set /can/ be ordered. The
Axiom of Choice for example implies that every set can be well-ordered
(which is a stronger statement than simply claiming a total order).

The point in dispute is your claim that there /must be/ some specific
"natural" order for /every/ infinite set, which is "inherent" in its
definition.

This is false; as my examples show: given two lines A and B in the
Euclidean plane, there is /no inherent order/ in the primitives,
definitions and axioms which /forces/ us to say A < B or A > B. Any
such ordering is a completely arbitrary structure "tacked on" /after/
the definition of L = "the set of all lines in the Euclidean plane".

(iii) This is a serious problem for your notions of "set size". If we
do not agree on the "natural" ordering of a countable infinite set,
then we will also in general /disagree/ on the "density measure" of
some proper subset of that set, "lim d(n)/n as n-> oo".

(iv) You seem to feel that the "natural" order of points in the
Cartesian plane is to order them lexicographically by their x and y
coordinates in R^2.

I prefer to order them lexicographically by their /polar/ coordinates
(theta, r); so that points near the origin come "before" points far
from the origin. I find this ordering more in keeping with the notion
of "distance".

So I have a "smallest" element (the origin); whereas you do not. And
every interval in my ordering is bounded; whereas you have many
unbounded intervals (e.g., the interval ((0,0), (1,1)) is unbounded in
either x or y, depending on your lexicographic ordering).

Suppose we want to compare the number of lines x = n for n an integer,
to the number of circles with integer radius centered at the origin.

If we use my ordering, the number of such lines contained in the
interval (0,0), (theta, r) is always 0; and the number of circles is
floor(r). So I find that that there are more circles than lines, in the
limit.

If we use your ordering, the number of such lines contained in the
interval (-x, -y), (x, y) is always 2*floor(x) + 1; and the number of
circles is floor(x). So you find that that there are more lines than
circles, in the limit.

Which of us is "right"? Which of these two orders is the one and only
"natural" one which is "inherent" in the definition of the Cartesian
plane?

Cheers - Chas

.

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