Re: Extending the reals

Tony Orlow wrote:
Okay, my bad. Still, if the ih-numbers are included, and lie between the
reals and 0, then the reals consist of three disjoint sets,
(-eta_1,0),0, and (0,eta_1). Since the hierarchy continues forever, each
such H_x would be disjoint from 0.


David R Tribble wrote:
You're still thinking of the suprareals as somehow being colinear
(on the same "line", within the same set, whatever) as the reals, and
they aren't. All of the reals are in H_0 (equivalent to R). All of
the suprareals based on eta_1 are in H_1. No member of H_1 is
a member of H_0, and vice versa. Any given suprareal h, which
is a polynomial with a largest term x*eta_j, is a member of H_j,
and not a member of any other suprareal (or real) set.


Tony Orlow wrote:
If that is the case, then there is no gap in H_1, since the reals that
consitute that gap are not in the set. Between any two members of H_1
lies another, no?

Except that the negative suprareals in H_1 (in H_1-) are not connected
to the positive suprareals in H_1 (in H_1+). There is no least
positive suprareal in H_1, nor is there a least negative in H_1.
Thus the negative and positive half-sets are not connected.
You can't get from say, -eta_1, to +eta_1 by successive incrementing
(or multiplying) by a positive number.

And in fact, H_1 is even more disconnected than that. The following
subsets of H_1 are not connected with each other:
x0 + eta_1
x0 + x1 eta_1
x0 + x1 eta_1^p1
x0 + x1 eta_1^p1 + x2 eta_1^p2
etc.


David R Tribble wrote:
All of the suprareal sets H_i are disconnected, meaning that
no two sets share any elements. That's what we mean by
"unconnected sets".


Tony Orlow wrote:
Only if you consider all H_x for x<i to be part of any H_i. But, you say
H_0 is not part of H_1, and so this is not the case. None of your H_1
have a gap if they are not colinear with some H_x "inside" of it which
is not included.

All the reals x are members of R, which is H_0.
All supranumbers based on eta_1, i.e., those of the form
x0 + sum{i = -n to +m} x_i eta_1^i, x_i in R
are members of H_1. Likewise, all suprarerals of the form
h0 + sum{i = -n to +m} h_i eta_1^i, h_i in H_1 U R
are members of H_2, and so forth. There is no subsetting
of H_i in H_j when i<j. All H_i are disjoint sets.


David R Tribble wrote:
However, you'll notice that my definition of H_0 (which is R)
includes 0 and 1. So yes, each H_i (other than H_0) is disjoint
from 0, i.e., does not include 0 as a member.


Tony Orlow wrote:
And what about your sets of inverses of eta_1? Does that include 0? It
would seem to me that 0 would be equal to H_-oo, so it's either included
in all higher H_i, or in none.

There is no suprareal h in any H_i such that 1/h = 0.
Just like there is no real x in R where 1/x = 0.
But 0 is in R, so it's in H_0, and it is the additive identity for the
suprareals, just like it is for the reals.

There is no set H_oo or H_-oo. There is a set H_x for every x in R,
but oo and -oo are not members of R.


David R Tribble wrote:
Since this is a hierarchy of unconnected/disjoint sets, it makes
no sense to talk about the interval (0, eta_1), because 0 is a
member of H_0 but eta_1 is a member of H_1, two disjoint
sets. You could say that they line on separate (non-touching)
number lines.


Tony Orlow wrote:
I would say they reside on uncountable distant countable intervals of
the real line.

Then you would be wrong. The term "line" implies a connected set
of points. H_i is a collection of unconnected sets.


Tony Orlow wrote:
0 is really eta_-oo.


David R Tribble wrote:
No, 0 is just plain 0, the additive identity for the suprareals and a
member of H_0.


Tony Orlow wrote:
Did you even think for a second about what eta_-oo would be?

No, because -oo is not a member of R, so there is no H_-oo.

I think having an uncountable number of H_x sets is enough to be
interesting for the time being.


David R Tribble wrote:
Any notion of "distance" for the suprareals will only make sense for
members of the same (connected) H_i set. Otherwise it's like
asking what the distance is between the members of {1, 2, 3}
and {red, blue, green}.


Tony Orlow wrote:
If you insist, but I don't understand why you do. What, again, does it
break for these numbers to all be colinear?

They can't be "colinear" because they are unconnected sets.
There is no way to start at, say, 2-eta_1 and start counting
(incrementing, adding, whatever you choose to call it) and end
up at, say, 3+eta_1^2 without somehow jumping from eta_1 to
eta_1^2. Which can't be done under the axioms I provide.

.

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: Extending the reals
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