Re: Does the Calculus rest on Euclid?

Nathan wrote:

Hatto von Aquitanien wrote:

If a real number is a "set of rationals", and there is only one real
number designated by the concept indicated by the phrase "a real number",
then, in the case of pi either the set called pi is identical to the
object called pi, or we are avoiding (or more correctly, evading) the
original topic.

It doesn't make sense to talk about what the object called pi "is".
Numbers don't exist in the same sense as the pen sitting on my desk.
Certain properties characterize pi, and especially its properties as a
member of the set of all real numbers. In different models of the real
numbers, pi might be a set of rational numbers, an equivalence class of
sequences of rational numbers, or even a point on a line. Whether or
not these objects are "identical" is irrelevant. The point is that all
these models behave equivalently in the ways that are significant for
real numbers. What color is pi? It doesn't matter.

Well, at the risk of posting on low blood caffeine, I believe it does
matter. Again, this is a topic to which I simply cannot direct my full
attention at present. What appears to be happening is a crossing of levels
of abstraction. This is how Russell tried to climb out of his paradox. My
instincts tell me that there is a contradiction to be found by reviewing
carefully all the statements and assumptions. Or perhaps there is a
necessary step missing which, if completed, would result in a
contradiction.

I am confident that at least one concept has been accepted from the
properties I ascribe to the continuum in the development of the real
numbers. That is the concept of completed infinity. OTOH, since
iterations are discrete, one can argue that they are countably infinite.

This really wasn't what I had in mind when I started this thread. I was
really trying to get a handle on a rather elementary idea in geometry.
That concept is the idea of measuring the length of the arc subtending a
radial angle in a unit circle. I may have been well advised not to have
mentioned the calculus.

I could see no way of going from the notion of an n-sided polygonal
"approximation" to the circle (or arc thereof) without invoking the Powers
on High. I was thinking in terms of Euclid, but my reasoning was following
the line typically used in arriving at the arclength of a curve. Since
virtually all developments I have seen for the arclength theorem invoke
geometric analogs, its hard for me to identify the which is "purely
analytical".
--
Nil conscire sibi
.

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