Re: An uncountable countable set

Tony Orlow wrote:
imaginatorium@xxxxxxxxxxxxx wrote:
Tony Orlow wrote:
Virgil wrote:
In article <454286e8@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:
Like, perhaps, the Finlayson Numbers? :)
Any set of numbers whose properties are known. Are the properties of
"Finlayson Numbers" known to anyone except Ross himself?
Uh, yeah, I think I understand what his numbers are. Perhaps you've seen
our recent exchange on the matter? They are discrete infinitesimals such
that the sequence of them within the unit interval maps to the naturals
or integers on the real line. Is that about right, Ross?

Do they form a field?

Brian Chandler
http://imaginatorium.org


Good question. Ross? What says you to this?

Here's what Wolfram says applies to fields:
http://mathworld.wolfram.com/FieldAxioms.html

My understanding, looking at each of these axioms, is that they apply to
this system, and that it's a field. I suppose you would want proof of
each such fact, but perhaps you could move the process along by
suggesting which of the ten axioms you think the Finlayson Numbers might
violate? After all, if you find only one, then you've proved your point.
Not that I am necessarily concerned with whether they form a ring or a
field or whatever, until that becomes important. Is it? Why the question?

Tony

It's a point of consideration that without some ball labelled infinity
the process doesn't complete, for the completion of the "supertask."

Consider Achilles and the tortoise, again, every distance between the
start and finish line is covered by each, and ten times as much by
Achilles.

Virgil is a troll, 'tis true.

The "Finlayson numbers" as coined by some other fellow, contain all the
numbers in the "Finlayson numerical model", named by somebody else, the
origin is (0, 0, 0, ....). That about sums them. The "Finlayson real
numbers", or as I generally call them the "real numbers", have
characteristics of being at once complete ordered field, and contiguous
points on a line. In terms of their scalar value, those "indefinite"
reals are defined as the immdiate neighbors, and "definite" reals
basically as Dedekind/Cauchy, which is insufficient (using Dedekind
cuts / Cauchy sequences, the "standard" method) to describe all real
numbers.

The real numbers are, macroscopically complete ordered field, and
microscopically partially ordered ring, and I see there being something
along the lines of a "rather restricted transfer principle", in the
words of Schmieden and Laugwitz, in terms of transitions or transitive
application of their form as predicates.

With a least positive real, for example as is illustrated in a
counterexample to standard real analysis, Dedekinf/Cauchy can't be
sufficient to describe a real number.

There is no set of numbers in ZF. There is no universe in ZF, that is
sufficient reason for many to abandon regularity and promote
alternative theories with alternative resolution of the Russell,
Cantor, and other "paradoxes" seen to result otherwise from
unrestricted comprehension. There are understandably less who would
say that regularity is a "false axiom", I do where there are only
primary objects in a pure object theory, saying that unrestricted
comprehension is natural.

The null axiom theory has as primary objects variously sets or numbers
and geometric forms.

There are only and everywhere real numbers between zero and one.

Ross

.

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