Re: Extending the reals
- From: cbrown@xxxxxxxxxxxxxxxxx
- Date: 31 Dec 2006 17:53:21 -0800
Tony Orlow wrote:
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.
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.
<snip>
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?
(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.
(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".
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)".
(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.
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?
(i) There is a difference between saying "A is a connected subset of
B", "A is a dense subset of B", and "A is the image of a continuous
function of R -> B". These concepts are related, but not synonymous.
(ii) In the suprareals, a subset Y is disconnected iff Y is the union
of two disjoint, open sets under the subspace topology of the
suprareals (which I am assuming is the topology of open intervals).
That is to say, for a subset Y of the suprareals to be disconnected,
there must be two (non-empty) sets Y' and Y'', such that:
Y = Y' u Y'', and
Y' n Y'' is empty; and
there are open sets U, V in the suprareals such that Y' = U n Y, Y'' =
V n Y.
(iii) The subspace topology on R has as its basis the intersection of
open intervals in the suprareals and the set R. Let r be any element of
R and e1 = 1/n1. Then (r - e1, r + e1) is an open interval in the
suprareals, and (r - e1, r + e1) n R = r.
Therefore every point in R is open in the subspace topology, and every
point is also closed in the subspace topology; so R gets the discrete
topology. Thus R is a totally disconnected subset of the suprareals. (I
erred when I previously stated that R inherits the usual topology).
(iv) In the suprareals R u L is disconnected. Let Y' = {x : x in R u L,
x > 0 and 1/x in R u L}. Y' is an open set (it is the union of open
intervals (x, x+1) for x in Y'). Let Y'' = {x : x in R u L and (x<=0 or
1/x not in R u L)}. Y'' is an open set (it is the union of open
intervals (x-1, x) for x in Y''). R u L = Y' u Y''; Y' n Y'' is empty,
and thus R u L is a disconnected subset of the suprareals.
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?
(i) Suprareals in H u R are polynomials. H is the set of suprareals of
degree 1 or greater. R is the set of suprareals of degree 0. A
polynomial of degree 1 or greater plus a polynomial of degree 0 is,
again, a polynomial of degree 1 or greater. So any element of H plus
any element of R is again an element of H.
(ii) A better (and generally less confusing) formulation of the
suprareals is to refer to them as the closure of <H, R> under
multiplication, division, addition and subtraction; which is to say
that they are rational expressions of the form p/q, where p and q are
polynomials over some set of eta_'s with coefficients in R; e.g:
x = (1 + 2*eta_1)/(3 - eta_1 + 4*eta_1^2)
What we have been calling "H" is then the set of all polynomials p of
degree 1 or greater (i.e., where the "denominator" polynomial is 1).
"R" is the set of all polynomials of degree 0. What we have been
calling "L" is the set of expressions of the form r + 1/p, where r is a
real, and p is a polynomial of degree 1 or greater.
This scheme does not account for all suprareals (as noted elsewhere).
For example, it does not contain p + r + 1/q for polynomials p and q
and real r. However, it can be written as
(p*q + r*q + 1)/(q^2)
which /is/ of the above form (ratio of polynomials).
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?
(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").
(ii) The numbers in H_R are (rational expressions of) polynomials in a
finite set of eta_'s with coefficients in R, with a finite number of
terms.
Say p and q are polynomials in H_R.
Then p is a polynomial over some finite set of eta_'s, call them eta_a,
eta_b, .., eta_m, where a, b, ..., m are real numbers; and having
coefficients in the real numbers, e.g:
p = 1 + pi*eta_a^2 - sqrt(2)*eta_b*eta_c + ... - 7*eta_m.
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.
Suppose r is a polynomial of the form eta_r, eta_s, eta_t, ..., eta_z.
Then p - q is a polynomial in eta_a, eta_b, eta_c, ..., eta_m, eta_r,
eta_s, ..., eta_z.
Since there are only a finite number of eta_'s, there is a "largest"
eta_x, where x >= a, b, c, ..., z /using the usual ordering of the
reals/.
Then we say p > q if the coefficient of the largest power of eta_x in p
- q is > 0.
To extend this to rational expressions, we say that p/q > r/s iff p*s -
q*r > 0 (with a few modifications to handle multiplication when q or s
< 0).
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?
(ii) Your statement makes no sense; because there is no such
"predecessor to pi" in the usual ordering of the reals.
(iii) Think of H_pi as being related to the set of all /finite/ subsets
of (0, pi] as a subset of R.
Each of these finite subsets then defines a finite set of eta_'s, over
which some set of polynomials is then defined over these eta_'s with
coefficients in R
H_pi is the union of all rational expressions of these polynomials,
taken over all finite subsets of (0,pi].
For example, {1, sqrt(2), 3} is a finite subset of (0, pi]. So
x = (1 + eta_(sqrt(2)) + (eta_3)^2)/(1 - eta_1)
is in H_pi; but
x = 1/eta_4
is not in H_pi, because 4 > pi, and thus 4 is not in (0, pi].
(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.
So what he has described is a total order on sets, which is /not/
synonymous with "a sequence of sets".
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.
(iv) You are.
Cheers - Chas
.
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