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

In article <45ac7d6c@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:

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.

One can give all sorts of models, i.e., systems satisfying the axioms.

Will one of those do?

In that case one can use the model of the Cartesian plane, RxR, with a
line being a set
L = {(x,y): a*x + b*y = c, a^2 + b^2 = 1; a,b,c,x,y e R} in RxR.



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?

That it doesn't help TO is TO's problem.


(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 set of lines tangent to a circle in the plane is infinite. But not
ordered.


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.

What "iteration" is required for the set of tangent lines to a circle?
.

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