Re: Extending the reals

David R Tribble wrote:
David R Tribble wrote:
Please read the article. My suprareals are not infinite numbers.

Tony Orlow wrote:
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.

Chas Brown wrote:
Except for the fact that the suprareals don't form a "line"; they are
toplogically disconnected. By "disconnected", I mean that the
suprareals are the union of two non-empty disjoint open sets.

The real number line is /not/ the union of two non-empty disjoint open
sets. Lines are connected; the suprareals are not connected; therefore
the suprareals are not a line.

Exactly. (I appreciate the answers, Chas.)
I thought my diagrams made this clear.



And yet, when you introduce the ih-numbers as a non-real halo around 0, then you have disconnected the positive reals from the negative reals, as the positive h-numbers are separated by the relatively infinitesimal expanse of standard reals between them. Where you include all standard reals, all ih-numbers, and all h-numbers, they form a single line, though in a sense it's a discontinuous line, since your system is restricted to finite polynomials, which are countable.

Tony Orlow wrote:
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?

Chas Brown wrote:
(i) "2^eta_1" is not a polynomial in eta_1; so "2^eta_1" is not
meaningful as a suprareal. It follows that "log_2(eta_1)" is not
meaningful for suprareals.

That follows from the h-numbers being restricted to polynomials over eta1, but that restriction itself seems to be simply declared to "appear" true, and I don't see why it follows. The fact that the inverse of a finite polynomial is not generally a finite polynomial indicates that this restriction makes some of the properties David desires in his system impossible. I am suggesting that this restriction might not be desirable. That's all.


(ii) eta_1 is not a cardinality; so it makes no sense to say "this set
has eta_1 elements"; just as it makes no sense to say "this set has
1/sqrt(2) elements".

eta_1 is not a fraction, but a number beyond all reals. While one could consider a non-integral such number, one would do better to assume eta_1 to be integral, with fractional differences covered by its finite real "halo".


One has at best "the cardinality of the suprareals <= eta_1 is the same
as the cardinality of the reals". Of course it is also true (as DT
proves) that "the cardinality of the suprareals > eta_1 is the same as
the cardinality of the reals"; and "the cardinality of the suprareals
<= eta_1 is the same as the cardinality of the suprareals <=
1/sqrt(2)".

Eh, cardinality. Do you want to be able to perform arithmetic on eta_1 or not?


(iii) There are as many suprareals as there are real numbers.
Therefore, you cannot in any case "list" the elements of the
suprareals, just as you cannot list the real numbers.

Also, it makes no sense to talk about "how many bits" could be
used to encode a given suprareal, since eta_1 cannot be represented
in a binary notation, since it's not a real. It's kind of like asking
how many bits are required to encode i or w (omega).



The imaginaries are an analog of the reals, and omega's not a number with any "value". If you have a number of elements, ceiling(log2) of that number is the number of bits required for the list. You can have infinite digital strings to the left, which represent infinite values, ala p-adics or T-riffics.

Tony Orlow wrote:
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?

Chas Brown wrote:
(i) It's not an uncountable sequence; because a sequence is always, by
definition, countable (that's the part that is "an idea that doesn't
make any sense").

Exactly.

Note that using this terminology, eta_b * eta_c is NOT EQUAL TO
eta_(b*c). It remains the polynomial expression eta_b * eta_c. The real
number labels a, b, c, etc. are only there to indicate an ordering.

Exactly.



You first define the successive etas as being the set of objects larger than all elements of the last. Of course, this means that each set includes its successor, and you no longer have a hierarchy. I mean, how do you have everything larger than everything larger than any finite? Isn't that an empty set? So, the successive notion ultimately doesn't make sense.

Then, you suggested an "uncountable hierarchy" using a continuous set to index the levels. That relies on the notion of some infinitesimal unit, or you don't have successive levels in your hierarchy.

Now, if you want to define an uncountable set of eta units, such that each is a real power of some standard eta_1, then anything above eta_0 is beyond the finites, and any real difference in the eta number will result in a value difference beyond any finite number. By introducing a particular formula, you can get a system that works. But, don't listen to me...

Tony Orlow wrote:
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?

Chas Brown wrote:
(ii) Your statement makes no sense; because there is no such
"predecessor to pi" in the usual ordering of the reals.

Exactly.



Then eta_pi is not a level in a hierarchy, is it?

(iii) It follows that if 0 < x < y where x and y are real numbers, then
H_x is a proper subset of H_y.

Or a proper superset, as I've said. If H_0 is all numbers greater than all ih-numbers, that includes the finites AND the h-humbers. So, H_0 would include H_1.


So what he has described is a total order on sets, which is /not/
synonymous with "a sequence of sets".

Exactly.

(Thanks, Chas.)


Yeah, thanks Chas.
.

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