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

In article <45a91c47$1@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> 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 '<'.

See Euclid, or better yet, Hilbert's revision of Euclid.

Lines are primitives and are not defined at all. And order of points on
a line comes considerably later.


Or:

The set of all triangles in the Euclidean plane.

Define "triangle" without "line".

Don't need to as lines do not require "<", See above.


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.

Neither lines nor triangles are inherently restricted to any particular
maximum number of dimensions.

TO is self-deluded by limiting himself to Cartesian geometries.
.

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