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

cbrown@xxxxxxxxxxxxxxxxx wrote:
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


To say there is no definition is hardly to satisfy a request for a definition. Hilbert's axioms treat points, lines and planes as "primitive", defining the relationships between them. But, if you want to talk about the set of all lines in the Euclidean plane, then you need to do a little better than that. Which of Hilbert's axioms do you intend to use to calculate the size of that set?

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.


Euclidean geometry does not address the question we're considering.

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?

In order to be able to express the set as a function of an independent variable, "iterations".


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.

Okay, then you simply say there are an uncountable set of points, and an uncountable set of pairs of points determining lines, so you have an uncountable set of lines. That's all very general.

You can make a distinction between Euclidean and Cartesian definitions of a line, but a line's a line. Unless Euclid offers some way to calculate the number of points in the plane, that distinction doesn't help much, does it?


(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).

Right. All I am saying is, to generate an infinite set, to define it at all, involves the use of SOME order. If you try to define what it means for a line to be part of your set, you're going to have to order the points to express how the unique lines are determined by them.


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.

No, I am saying two different things, First, in order to explicitly define any infinite set one must employ some notion of order, whether in terms of intermediate values or successors, which associates the generation of a particular element with a particular iteration of element generation. Second, when dealing with subsets of the reals there is a common quantitative order, which can be employed to compare the sets over any given value range, including the entire real line. Those are two different statements. I am not saying that every set has one correct order, but that an ordering on a set can be used on its subsets to provide a common yard stick between them.


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".

How do you distinguish one line from another? If they are elements of a set, they must be distinguished from each other, but in the Euclidean plane, they have no particular identity.


(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".

Yes, which is why your definition of the set doesn't lend itself to any better estimation than "uncountable". If there is a generating process with an independent variable counting iterations, it's a different story, right?


(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".

Hmmm, well those aren't Cartesian coordinates, but that's okay. If you want points close to the origin to come first, each point should be the pair {r,theta}.


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).

Why do I need a smallest element? I just need to be able to compare any two.

We were talking about lines, and now you're talking about intervals. I don't mind unbounded intervals.


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.

Since each vertical line is tangent to exactly one circle and each circle to exactly two verticals, I'd say there are twice as many verticals as circles. That doesn't require any ordering, really, since it's a linear relationship.


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.

You mean completely contained within r of the origin? Of course not, but the portion of the line including the point of tangency is contained, and the correspondence apparent.

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.

I guess the circles would be floor(x)+1, including the circle of radius 0 at the origin. Forgot about that one. Yes, there are more lines than circles. In any case, none of the vertical lines is completely contained in that interval either, so I don't see the distinction here.


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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