Re: Non-Standard Analysis & Other Infinities

>> 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

.

Relevant Pages

  • Re: Non-Standard Analysis & Other Infinities
    ... Jon, that's not crankish talk. ... considerations of mathematical considerations pure and applied. ... Iota is a least positive real. ... The reals are a complete ordered field. ...
    (sci.math)
  • Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
    ... > In the normal ordering, what is the least member of the set of all ... Do you mean the least element of the set of all reals greater than ... non-standard considerations of the real numbers and the number system ...
    (sci.logic)
  • Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
    ... > In the normal ordering, what is the least member of the set of all ... Do you mean the least element of the set of all reals greater than ... non-standard considerations of the real numbers and the number system ...
    (sci.math)
  • 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)