Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 01/30/05
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Date: 30 Jan 2005 05:40:12 -0800
Hi Tim,
That's interesting.
I just reiterate.
About the use of a definition of computable, there does appear to be
some leeway in the matter. Everyone reading this is for the most part
comfortable with using the term equivalent to mean equipotent or
equipotent in that sense thus that they're interchangeable.
About Cantor's first and the ordinal index of the well-ordering, if
there only needs be one infinite ordinal, N, appended to the natural
index of a soi-disant well-ordering, then 1: that's not an uncountable
set of ordinals, and 2: it would imply in the well-ordering two points
of the reals between which there are not other reals.
The sole missing element in a bijection between set and its powerset is
often the case, leading to "infinity equals zero". For example in the
powerset result for f(x)=x+1, S = {}, f(N) = N+1 = P(N), N E P(N), with
ubiquitous ordinals. N is a limit ordinal.
I have much to learn about this transfer principle notion, these are
discussions about the real numbers.
Let's consider iota some more: for any 1/n, it's smaller, yet it's
greater than zero. I call the consideration of iota an "inductive
impasse", for there are two chains of induction that would block and
overflow each other, or just one that stumbles over itself. On the one
hand for 1/n there exists 1/(n+1), yet for that there exists 1/(n+2),
which is simple, obvious, and unfulfilling. Perhaps an alternate way
to visualize something that fits the properties of iota is that the
positive reals are unbounded and yet dense. Completeness means they
have the gaplessness property.
That gets back into cyclical definitions which is why Euclid defines
points and lines in terms of each other. A structure of contiguous
(and overlapping, one-sided) points that comprise a line are anonymous
with regards to their scalar value, yet there's nothing else those
points on the line could be. With only the knowledge that these points
comprise the line and that the line or rather segment or ray represents
the continuum (of real numbers), then one of those points is the least,
by the usual ordering of points on a line, in any non-empty collection,
or set, of those points.
Besides academic novelty and possible requirement for well-ordering the
reals and other reasons of mathematical necessity per allowance of
non-contradictory things, here's one reason I consider this structure
of interest: because it can help explain EF, the Equivalency Function,
which apparently nobody understands. (Except me, hee hee).
The issue of universal provability, or "a contradiction entails no
truth", is something where beneath axiomatizable set theory the
ur-element is similar to an unobservable particle in physics, yet it's
not. It suits what ails you, it's a cure for your needs. The
ur-element, the Ding-an-Sich or Thing-in-Itself, the Being and Nothing
and Universe and Void, the context about itself, is very mutable stuff,
and abscence of it. It's a set only by virtue of being in a set
theory, but it's everything else a set can't be. It's the ur-paradox,
but it can't be. Reading about the Ding-an-Sich and Being and Nothing,
philosophical notions, actually does help to understand. Buddhist
monks just stare at a wall for seven years, that's considered a path to
enlightenment. Truth entails all truth.
Mathematical logic is supposed to be about mathematics and logic, so
from the ur-element draw out some integers. They're conveniently Z and
N at the same time, the cumulative hierarchy, and happen to be all the
sets at once.
"Infinitesimals have infinitesimals". (Class of all classes is a noun
and causes set theorists twitches in their sleep.) What axioms do you
think are important, and why? I just say none of them are important
because all the non-logical or proper axioms except regularity are
theorems of first-order logic by itself with an ur-element that is
dually minimal and maximal, and tautology.
Yep, just more of the same.
Regards,
Ross Finlayson
- Next message: Helene.Boucher_at_wanadoo.fr: "Re: Name the thesis: "Formal sentences capture informal ones""
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