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David R Tribble wrote:
David R Tribble wrote:
Over the last few months I've been noodling around with the concept
of an extension to the reals, defining real-like numbers that are
larger than any regular real.
Chas Brown wrote:
Nice job. In conversations with Tony Orlow, I also thought of a similar
system (essentially, an ordered field extension of R using a polynomial
basis over some "unit infinity" B).
David R Tribble wrote:
I was hoping this conversation would go quite a bit further
before Tony's name was mentioned. His "unit infinity" is an
extermely flawed and inconsistent concept from the get go.
Oh well.
Tony Orlow wrote:
Did you honestly think that was a reasonable expectation? You seem to
have agreed that "larger than any finite" is a reasonable definition for
"infinite" That's a good start, a unit infinity. Welcome to the club.
Please read the article. My suprareals are not infinite numbers.
I have now read it. Yes, you are very careful not to call them
"infinite", and even to point out that, even though you illustrate them
as residing colinear with the reals, they are not really in that
relationship. I didn't see the point in tiptoeing around that,
personally. Your h-numbers are "larger than any finite", meaning farther
along the line from 0.
They are numbers that are simply larger than the standard reals.
In most other respects, they act just like reals. See Theorem I
(infinity) in particular:
-oo < h < +oo for all h in H.
Yes, I see that you made a distinction between absolute oo and the
h-numbers. I agree with that, just like infinitesimals are not absolute 0.
David R Tribble wrote:
For the record, I was inspired by some concepts of NSA for
creating my h-numbers.
Ross A. Finlayson wrote:
Your "seed", or eta_1 or what have you of your h-numbers, does look
quite similar to Tony's H-riffics' "unit infinity", or Yaroslav
Sergeyev's grossone or (1), or the unit scalar infinity. However, it
appears they are more similar to the Robinso(h)nian hyperreals. Tony's
rule would seem to apply, or not.
David R Tribble wrote:
They have nothing in common. My n1 (eta_1) is in almost all respects
another kind of real, obeying relations such as 1+n1>n1. Infinite
numbers don't act that way, and I make it quite clear that n1 is no
such thing. They are similar to Robinson's illimited numbers, and
I'd like to know just how closely related.
Tony Orlow wrote:
David's actually right. Despite the fact that he used "h" for his
numbers, after considering "t", his numbers correspond to the T-riffics
rather than the H-riffics,
No they don't. "t" stands for something else, and I leave it to you
to guess what. "h" originally was supposed to mean "huge" or "hyper",
but "hyperreal" has already been used, so now I'm settling on
"suprareal". "H" is a capital "eta", from my seed constant eta_1 (n1).
I chose eta because it looks like "n", for "number".
I may end up changing the notation anyway.
Okay, just as long as you don't start calling them Big'Uns and Lil'Uns.
Them's mines. ;)
... which are based on nested powers of 2 to
enumerate the reals. The T-riffics are a simple extension of the normal
digital numbers system, based on sums of powers of 2, or any number
base, preferably prime. They also correspond to Robinson's illimited
numbers, uncountably distant from each other, with countable rational
neighborhoods of digits surrounding limit points.
I don't think you understand the hyperreals of NSA.
My suprareals resemble the hyperreals, or at least the suprareals
in H1 and L1. I'm not sure how they're related, and I don't think the
hyperreals are related to the rest of the suprareal hierachy (H2, L1,
H3, L3, etc.).
Well, at least we both agree with Robinson that there's no smalleest
infinity in such a system.
In any case, your
"new" idea is not much different, at first glance, from my old and
"extremely flawed" concept. I'll print out and peruse your theory, and
then give more detailed comments, okay? Good job!
It's completely different. For one, it uses axioms to define the
existence and properties of the suprareals. You don't have any.
Secondly, your "unit infinity" is supposed to be equal to the
cardinality of both [0,1] and N, which is inconsistent within standard
arithmetic. And there are more flaws, which have been pointed
out to you many times.
Big'un is the number of reals in (0,1] and the number of elements in *N,
the hypernaturals, both uncountable. I've made that clear multiple times.
But please, this thread is supposed to be about my theory, not
yours. I'll be happy to join the thread you start for yours.
Okay, just noting similarities.
Ross A. Finlayson wrote:
Are you able to formalize otherwise inaccessible or what are
deemed paradoxical true results using your system?
Such as?
Tony Orlow wrote:
Oh, can you resolve the Continuum Hypothesis, or bring the subset
relation in line with infinite set size measures? Can you resolve the
difference between Galilean models of bijection and notions of set
density? You know, stuff like that.....
Those have all been resolved, regardless of your opinions about them.
The CH has been shown to be unresolvable within ZFC, so I would
speculate that the only way to "resolve" it would be to accept CH or
~CH as an additional axiom (which seems to be the usual approach)
or to try to add new axioms that force CH or ~CH.
At any rate, my little additional axioms won't do that.
Oh well.
I had a few comments:
1. In the section "Even More Numbers," you say, "In fact it would appear
that every h-number can be represented as a polynomial over powers of
eta_1." Is that to say that one cannot have log_2(eta_1) and produce
another h-number using that function? How many bits are required to list
eta_1 elements?
2. When you speak of the h-numbers as being disconnected, with the
intervening set of standard reals between their negative and positive
elements, does it not occur to you that including the ih-numbers creates
the exact same situation for the standard reals, such that the positives
and negatives always have something between them? Does this make the
reals not a continuous set?
3. At the end of "Still Bigger Sets", you say, "Every element in this
set is either a real or an h-numbers (sic), an ih-number, a real plus an
ih-number, or an h-number plus an ih-number." Can it not also be a real
plus an h-number, or even a real plus an h-number plus an ih-number?
4. "An Uncountable Hierarchy" struck me as odd, coming from you, David.
On the one hand, you are enumerating a sequence of sets, each defined as
being the elements larger than all elements in the previous set (rather
like limit ordinals) but then you suggest that each set may be numbered
with a real. Are you suggesting an uncountable sequence of sets, which
you previously proclaimed to be an idea without any sense? What is the
set H_pi the set of, all elements larger than the element of H_x, where
x is the predecessor to pi in the natural order of the reals? I can see
how this might be defined, if each H_x uses an eta_x which is equal to
eta_1 to the x power, but I didn't see that defined there, and I imagine
you don't want to get that specific about the measure of the etas. I
could be wrong.
Anyways, that's my comments. Happy thinking!
Tony
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