Re: Extending the reals

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.


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.

(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.

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).


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.


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.


(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".

Exactly.

(Thanks, Chas.)

.

Relevant Pages

  • Re: Extending the reals
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  • Re: Extending the reals
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  • Re: Extending the reals
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  • Re: Extending the reals
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  • Re: Calculus XOR Probability
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