Re: Non-Standard Analysis & Other Infinities

Jonathan Hoyle wrote:
> >> Jon, that's not crankish talk. Consider the dichotomy of
> >> points and line segments, and how that is a staple of
> >> mathematical discussions since antiquity. Today there are
> >> frameworks of modern discussion of these issues, and they
> >> lay at the root of very meaningful considerations of
> >> mathematical considerations pure and applied.
>
> Viewing points on a line in the same way Euclid did I have no problem
> with. The issue is your inserted assumption that the points are
> sequential. This is a violation of the very principle you are
> suggesting, since even from antiquity they knew that between any two
> distinct points there is a third in between them.
>
> >> Iota is a least positive real.
>
> There is no least possible real. As stated before: if iota is any real
> (or hyper-real), then iota / 2 exists and is less than iota.
>
> >> It's basically consideration that if the line segment is comprised
> >> of points, and each point on the line segment can only border
> >> at most two other points, then those points are its neighbors in
> >> a contiguous sequence.
>
> Your consideration is incorrect. It is true that the line segment is
> comprised of points. It is incorrect to say that each point on the
> line segment borders ar most two other points. As is true with the
> real line, it is also true with the geometric line segment that between
> any two distinct points, there lies another between them.
>
> >> The reals are a complete ordered field.
>
> Correct.
>
> >> For reals x, y, (x+y)/2 is a distinct number.
>
> Correct again.
>
> >> I think it's necessary to consider in the dichotomy that
> >> those distinct reals are distinct in a manner that applies
> >> to their distinction in the complete ordered field, yet at
> >> the same time it is necessary to consider all possible
> >> aspects of the reals, so there is a justified notion that the
> >> reals are as well a contiguous sequence of points...
> <snip>
>
> None of the rest of this makes sense to me. I don't know what
> "dichotomy" you are talking about. The points on a line are not
> sequential (whether it's Euclidean or the real line). Your constant
> presumption that they are is provably false.
>
> >> Consider the variation of Cantor's first as applied to
> >> well-orderings of the reals. Are the reals not a set...
>
> The reals are a set.
>
> >> ... or is choice inconsistent with ZF...
>
> Choice is provably independent to ZF.
>
> >> ...or are there uncountably many nested intervals, and
> >> so on and so forth?
>
> There are only countably many open disjoint sets on R at any one time.
> In your construction, your intervals are not open disjoint sets.
>
> >> Prior to Robinson with his hyperreals, there were a
> >> variety of considerations of post-Weierstrass "nonstandard"
> >> rationalizations of the real numbers. There are as well
> >> also today others.
> <remaining post snipped>
>
> I don't know what you are referring to here. But whatever it is, it is
> not related to the discussion between Dave & Daryl.
>
> Jonathan Hoyle
> Eastman Kodak

Hi,

First, about the nested intervals, they are non-degenerate closed
intervals in the complete ordered field. The construction may as well
have the nested interval endpoints indicate open intervals, or
half-open intervals, as they are contained by nested, non-degenerate
closed intervals. So, I still don't see the notion about the intervals
being closed as affecting the progression of the argument, that a
well-ordering, which as described over in "Well-Ordering the Reals" is
a bijection between R and some ordinal O', has the same consequences as
R bijecting to an ordinal N.

If otherwise, please explain what you mean more fully for those reading
our current conversation who are not necessarily up to speed on that
variation of Cantor's first or nested intervals. I think above Daryl
was talking about diagonalization instead of nested intervals, also
known as Cantor's first proof the uncountability of the reals, or as I
hope to show Cantor's first proof of sequential points in the reals.
Well-order the reals.

R is a complete ordered field except generally division by zero is
undefined. It has no zero divisors, which is a different issue that
that for no x, y =/= 0, xy = 0. Various considerations of division by
zero lead to for example meromorphic functions as I heard about a while
back in "f(f(f(y)))" and projectively extended real numbers with a one
or two point compactification. ("You're dividing by zero!")

In talking about the hyperreals and various other nonstandard
constructions of the reals, you may as well consider for further
comprehension notions a la Schmieden and Laugwitz' dually partially
ordered ring and complete ordered field reals, which predate Robinson's
construction a few years.

There are everywhere and only reals between zero and one.

Ross

.

Relevant Pages

  • Re: Well Ordering the Reals
    ... >>>an initial segment indexed by the natural integers. ... If we let c be the cardinality of the reals, ... It will be some infinite ordinal ... > segments of the well-ordering, else there would be countably many. ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... >>In a well-ordering of the reals, ... >>an initial segment indexed by the natural integers. ... For each countable initial segment is generated a unique convergent ...
    (sci.math)
  • Banach-Tarski: Staccato
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  • Re: Proper Turing cones are null?
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    (sci.logic)
  • Re: Help. What is a model?
    ... explaining why I do, or would, avoid the phrase. ... But unless you can define what 'true' means then you haven't defined the interpretation function, ... treatments, but as I said, ordinarily, reals are not defined as ... complete ordered field, that set is uncountable. ...
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