Re: Rational numbers, irrational numbers: each dense in real numbers

On Sep 18, 12:27 pm, finite guy <adamle...@xxxxxxxxxxxx> wrote:
It's simply why you can't have circles, spheres or cubes.

OK, it seems as if there have been many so-called
"cranks" who have appealed to geometry in order
to justify their claims regarding set theory. And
there has been a wide variety of views on this, from
RF's belief that a line segment contains only
countably many points -- and that points may have
successors on the line -- to WM's "set geometry,"
and now finite guy's belief that circles and spheres
cannot exist.

Let us consult both Euclid's original postulates, and
the modern postulates given by Hilbert, and see how
the so-called "cranks" and their geometries measure
up to Euclid and Hilbert:

http://www.friesian.com/space.htm

We begin with RF and his claim that points on a line
segment may have successors -- infinitesimals, to
be sure. Several posters have made such a claim,
and the usual response is that it would contradict
Euclid's First Postulate:

First Postulate: To draw a line from any point to any point.

To me, this is a bit vague. Was Euclid merely saying
for every pair of points, there exists a line that
contains both points? Or was Euclid implying that
_between_ every two points (on the line that
contains them, of course) there must exist yet
another point?

If the former, then not only may points have
adjacent successors, but even _finite_ geometries
are not excluded:

http://en.wikipedia.org/wiki/Finite_geometry

Even in the latter case, clearly if between every two
points there exists another point, then a line must
be dense and consist of infinitely many points. But
still, that doesn't imply that a line must have
_uncountably_ many points. (Of course not, since
uncountability didn't exist in Euclid's day.) Indeed,
recall that Euclid dealt mainly with constructions. I
fail to see how any of Euclid's axioms imply the
existence of any segment whose length is not a
constructible number, such as cbrt(2) (Doubling the
Cube) or pi (Squaring the Circle). So Euclid's
Axioms imply the existence of only countably
many points.

(Before we leave Euclid, one can't help but wonder
about Euclid's Fifth Axiom -- that's _Axiom_, not his
Fifth _Postulate_:

Fifth Axiom: The whole is greater than the part.

which seems to imply the nonexistence of
Dedekind infinite sets, or even the Third Axiom:

Third Axiom: If equals be subtracted from equals, the remainders are
equal.

applied to Dedekind infinite sets. It's interesting
how no one seems to bring that one up!)

Of course, the defects in Euclid are supposed to
be remedied in Hilbert. Returning to the link:

http://www.friesian.com/space.htm

we see that if we only admitted the axioms
labeled "Axioms of Incidence," it is obvious that
a finite model exists. The "Axioms of Order" do
prove that infinitely many points exist. And
finally, Archimedes' Axiom implies that no
non-Archimedean segments (infinitesimals)
exist, so RF geometry is no longer a model.

On the surface, the Axiom of Line Completeness
appears to imply that there exist uncountably
many points (just like the complete metric
space R):

(Axiom of Line Completeness) An extension of a
set of points on a line with its order and congruence
relations that would preserve the relations existing
among the original elements as well as the
fundamental properties of line order and congruence
that follow from Axioms I-III, and from V,1 is impossible.

In other words, if L is a line that satifies the
previous axioms, then one can't add points
to L and still satisfy the axioms. It's not
obvious to me how this must imply that there
are uncountably many points. Indeed, since the
other axioms imply only the existence of
segments whose lengths are constructable, one
could argue that the Axiom of Line Completeness
prevents us from adding points to a line whose
distance from each other is not constructable
(like cbrt(2) and pi).

So where does one get the idea that there exist
uncountably many points anyway? When I took
high school geometry, the very first postulate in
the book is the Ruler Postulate:

http://www.mnstate.edu/peil/geometry/C2EuclidNonEuclid/3Ruler.htm

The Ruler Postulate clearly states that there
exists a bijection between the set of real numbers
and the set of points on a line, and since the
former is uncountable, so is the latter. But the
Ruler Postulate is not listed among the axioms of
either Euclid or Hilbert, unless the latter's Axiom
of Line Completeness is intended to be
equivalent to the Ruler Postulate.

Perhaps I'm wrong, and I'd like to see how exactly
one implies the existence of uncountably many
points (not real numbers, but _points_) using only
Euclid or Hilbert (not the Ruler Postulate, unless,
once again, Line Completeness is intended to be
its equivalent).

What about finite guy? FG's claim that no circles
exist clearly contradicts Euclid's Third Postulate,
yet Hilbert says nothing about circles. In the
model of geometry in which only constructable
points exist, circles may exist, but a segment of
measure pi doesn't (Squaring the Circle). FG also
tells us that cubes don't exist, and he presents
Fermat's Last Theorem as his "justification," but
FLT has nothing to do with geometry.

.

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