Re: Extending the reals

Tony Orlow wrote:
David R Tribble wrote:
Chas Brown wrote:
The real number line is /not/ the union of two non-empty disjoint open
sets. Lines are connected; the suprareals are not connected; therefore

Tony Orlow wrote:
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,

Chas Brown wrote:
It turns out that not just the positive and negative reals are
disconnected in the suprareals; in fact, /every/ two disjoint sets R1
and R2 with R = R1 union R2 are disconnected (this is what "totally
disconnected" means).

I think what you mean to say is that they form a total order (sometimes
misleadingly called a linear order). The real line is one /example/ of
a total order; but there are many, many other types of total orders,
sometimes with quite bizarre properties which make them very much
"unlike" the real line.

Right. Introducing the i-suprareals (infinitesimals) does not make the
existing reals unconnected. Think of the suprareals and i-suprareals
as sets of "numbers" separate from the set of reals.

Which means that it's not proper to think of the union of the reals
and the suprareals as a single "number line", but more as separate
disconnected number lines. More properly, they are disconnected
sets.



Then thesame applies to theh-numbers. If they are not colinear with the
reals, then the reals do not constitute a "gap" within the suprareals.
If you apply the logic of connectedness to the suprareals as you do to
the reals, then they are also gapless. For any two suprareals, positive
or negative, there lies a suprareal between them, Therefore, they are as
"continuous" as the reals.


For the third time in this thread, "dense", "connected", and
"continuous" have different meanings. And "gapless" has no particular
meaning to me at all; unless it is supposed to mean "dense".

(i) The fact that the suprareals are a dense set does not mean that the
suprareals are connected.

(ii) The fact that the suprareals are a dense set does not mean that
there is a continuous function from the reals onto the suprareals.

These two statements follow from the definitions of "dense",
"connected" and "continuous".

http://en.wikipedia.org/wiki/Dense_set
http://en.wikipedia.org/wiki/Connected_set
http://en.wikipedia.org/wiki/Continuous_function

Or better yet, read a book on topology: Hocking and Young's "Topology"
is about $13, and they probably have it at Barnes and Nobles.

http://www.amazon.com/Topology-John-G-Hocking/dp/0486656764

Tony Orlow wrote:
though in a sense it's a discontinuous line, since your system is
restricted to finite polynomials, which are countable.

Chas Brown wrote:
No. Since the coefficients of the polynomials in question are real
numbers, there are, for example, an uncountable number of polynomials
of the form "a + b*eta_1".

BTW, the "reason" why the suprareals are not connected is not because
there are "too few" suprareals; it's because there are "too damned
many".

This is an example of where just relying on your "gut" can lead you
astray. Your "gut" may say that all we need to do is "insert" more
"suprareals" until "all the gaps are filled up", and then it will be
connected.

But that's what we just /did/ - we "inserted" a bunch of numbers to
"fill up" the reals to make the suprareals, and yet the result is not
"more connected", it's actually much /less/ connected.

Exactly. Even the simplest construction, x+eta_1 for any real x,
produces an uncountable set.

What is curious (and which I will probably add to the next revision)
is the fact that
x0 + eta_1
x1 eta_1^1
x2 eta_1^2
...
are unconnected (uncountable) sets. Any member of one set in the
list is less than any member of the next set (assuming all positive
real x's).



That was the suggestion I offered: define you H_x as the countable
neighborhood of eta_1^x, whether x is natural or real.

(i) What is a "countable neighborhood"? A neighborhood is usually
equivalent to "an open set", which in this topology means "an open
interval in the suprareals". But what is a "countable neighborhood"?
And how do you define an open interval prior to defining a total order
on the elements you aim to define?

(ii) Why do you say "/the/ countable neighborhood"? Why is there only
one unique such object associated with eta_1^x?

(iii) How does your definition of "countable neighborhood" /define/
H_x; by which I mean provide a /definition/ of addition, subtraction
and multiplication on the elements of H_x?

Any real
difference in an exponent applied to an infinite value results in an
infinite difference, an uncountably "disconnected" pair of sets.


(i) How does a real difference in the exponents x, y of eta_1 get
"applied" to an "infinite value" to "result" in an "infinite
difference"?

(ii) What is the difference between a "disconnected" set and an
"uncountably disconnected set"?

(iii) What is an "uncountably "disconnected" pair of sets"? What two
sets of suprareals are you referring to?

(iv) Why does it follow from these definitions that "an infinite
difference" is therefore "an uncountably "disconnected" pair of sets"?

Yes. I have two choices:
a) allow suprareals to be polynomials with an infinite number of
terms;
b) accept that H_1 U L_1 is not a field.

Option (b) appears to be the more acceptable at this point, because
it's simpler and because it allows me to keep the more general
suprareal construction as being a polynomial with both positive and
negative integer powers of eta_i.

If that's the construction, then I don't see how you can form your
"uncountable hierarchy", since there aren't an uncountable number of
integers. That's what I was saying about that part not jibing.


Let Eta = {eta_x : x in R+}. Let A = {a, b, c, ..., m} be a finite set
of reals. Let E be the set of eta_'s defined by E = {eta_u : u in A}.

Let h = p(E) be a polynomial with real coefficients over the set
{eta_a, eta_b, eta_c, ..., eta_m}.

Then if y is a real number greater than any real number in the (finite)
set A, then we /define/ eta_y > h.

Given that definition, can you work out whether

1 + (eta_2)^100*(eta_3)^2 > 3 - 2*eta_e + 17*eta_pi

is true or false?

If not, can you instead first work out whether

eta_pi < 1/17*(-2 + (eta_2)^100*(eta_3)^2 + 2*eta_e)

is true or false?

And the suprareals can be compared to reals, although I'm not sure
that's what Tony meant by "beyond". Again, the suprareals and the
reals do not reside within the same "number line" or connected set.



But, did start by assuming some number beyond the reals, such that
eta_1>x for all x in R, yes?


If by "eta_1 is beyond the reals", you mean the assertion "in the
suprareals, r < eta_1 for all r in H_1 such that r is of the form (r_0,
0, 0, ...)", then yes. If you mean something else, then I can't opine
at this point.

Tony Orlow wrote:
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".

Chas Brown wrote:
One would do better to be specific about what one means, rather than
sling around mathematical sounding phrases without specific meaning.
"Fractional differences covered by its finite real "halo" " is hardly
comprehensible to me.


I don't think that's my problem, and you needn't get snippy about it. A
simple request for clarification will do.


OK. Given /David's definition/ of H_1:

(i) What does it mean for eta_1 to be "non-integral" in the suprareals?
What does it mean for eta_1 to be "integral" in the suprareals?

(ii) What are "fractional differences", in the context of the
suprareals? Are the "fractional differences" of eta_1 the same as, or
different from, the "fractional differences" of any other suprareal?

(iii) What is the "finite real halo" of eta_1 in the suprareals?

(iv) What does it mean to say "the fractional differences of eta_1 are
covered by its finite real halo" in the suprareals?

(v) Assuming that we can now make sense of your assertion, why would it
be /better/ to "assume" eta_1 to be integral, with fractional
differences covered by its finite real "halo", instead of "merely"
assuming that eta_1 is "non-integral"? What, exactly, is "better" about
it?

The question is whether eta_1 can be used as a count, or set size. It
cannot, if it's not a whole number.

As I use the term "the number of elements in set X = card(X)", it
cannot, if it is not a /cardinal number/.

If eta_1 is "the number of elements in the set X", it means to me that
there is a bijective function f : eta_1 -> X.

But the /suprareal/ 2*eta_1 is /not a set/; except in the trivial sense
that the sequence (0, 2, 0, 0, 0,...) is a set.

So to say "the set Y has 2*eta_1 elements" no longer makes sense to me;
unless you mean that there is a bijective function between the set (0,
2, 0, ...) and set Y (which would be an odd thing to mean, since then
there also exists a bijection between the suprareal 0 = (0, 0, ...) and
Y).

I have no real idea of what you mean by "eta_1 is the number of
elements in set X", if you /don't/ mean "the cardinality of X".

Consider it a whole number

What is a "whole number", when we are not talking about the reals? In
particular, what is a "whole number" in David's H_1?

and any
non-integral suprareal as requiring the addition of a finite real
component. The "halo" in NSA is the countable neighborhood of
infinitesimals that can be considered to lie around each real number.

In
similar fashion, we can view numbers like the suprareals, which are
separated by an uncountable number of units from each other, to have
countable neighborhoods of unit intervals surrounding them, each exactly
the same as the standard real line. That's what I mean by a "halo". It's
the same thing on a different scale. Questions?


(i) What does it mean to say "suprareals x and y are separated by an
uncountable number of units"?

(ii) What does it mean for a suprareal y to "be considered to lie
around" the suprareal x = 1 + eta_1^2?

(iii) From these definitions, why does it follow that there is only a
countable number of such infinitesimal y's? Why isn't there an
uncountable number of such infinitesimal y's?

(iv) What does it mean for a neighborhood in the suprareals to be "a
countable neighborhood of a unit interval"?

(v) How does a collection of "countable neighborhoods of unit
intervals" then "surround" a suprareal x? When do they /not/ "surround"
some other suprareal, y?

(v) What does it mean for a countable neighborhood of a unit interval
in the suprareals to be "exactly the same as" the real line?

(vi) When you say "that's what (you) mean by a halo", what is "that"?
Is "that" a "way of viewing" the suprareals? Is "that" some particular
set of suprareals? Is "that" a property of certain sets of suprareals?

Etc.

If I asked you what you meant every time you made an assertion, these
threads would get even longer. It's usually better to ignore your
statements unless I can make /some/ sense of them.

That's why you should /learn/ some of the terminology. Then we can
understand each other much better!

We are not considering "numbers" here in the usual sense. We are
considering mathematical objects which share many, but not all,
properties of what are usually called "numbers". We can define
something that acts very much like addition between the usual numbers;
so much so that we call it "adding two suprareals"; but it is important
to keep track of the fact that they are /not/ numbers as we usually
think of them - they are "abstract numbers".

Um, what is the definition of "number", please?


There is no "one definition"; it depends on /context/.

For the H_1, a "number" means a sequence of real numbers x = (a_0, a_1,
...., a_n, 0, 0, 0, ...) where for some natural n, m > n imples a_m = 0.
"+" is a commutative group operation defined as pointwise addition;
multiplication is a little more complicated to describe. The result is
a (commutative) ordered ring.

For the hyperreals, a "number" means an /equivalence class/ of
sequences of reals, where sequences a and b are considered equivalent
if they agree on some element of the chosen ultrafilter. Both addition
and multiplication are defined point-wise; and the result is an ordered
field.

For cardinal numbers, "numbers" is a particular a subset of the
ordinals. Adiition and multiplication are defined, but the result is
not a ring (neither + nor * are groups; they are monoids) (although it
is totally ordered).

For the complex numbers, "numbers" are expressions of the form "a +
b*i", where and b are real numbers. The complex numbers form a field;
but /not/ an ordered field: it is not the case that, for any two
complex numbers u, v, that exactly one of u<v, u>v, or u=v must hold

For algebraic numbers, "numbers" are a subset of the complex numbers a
in A that satisfy P(a) = 0 for some polynomial P in Z[x]. + and * are
defined so that the result is a field.

For a prime number p, the field F_p^n consists of p^n "numbers" which
form a field. It is not ordered.

Etc.

In general, "numbers" is usually shorthand for "the elements of some
particular set X, with operations + and * defined so that (X, +, *)
forms a commutative monoid with multiplication"; but even that is
probably best considered just a guideline.


As such, eta_1 is not a "quantity" in the sense that "5" is a
"quantity". And it's not a "measure" in the sense that we say "the
length of the diagonal of the unit square is sqrt(2)".

It's a symbol which represents an abstraction.

Yes, well, symbols are all very well and good, but they are not all
there is to math.


And yet in this case, it is all that is being claimed.


Furthermore, eta_1
a. is not a real
b. is not a fraction
c. is not a sum of reals
d. is not prime
e. is not composite

I really think you might want to give more thought at some point to
those last two. There are reasons to consider each possibility.


Properties (a) and (c) mean that there is no possible decimal or binary
representation for eta_1.

That is not correct. Eta_1 CAN be treated like the T-riffic Big'un, but
there's no point arguing that at this point.


Properties (d) and (e) apply to eta_1 the same way that they apply to
1 or i.


What "way" is that?

By the definition of the complex numbers, every complex number divides
i; just as every complex number divides every other complex number.
"Primeness" and "compositeness" make no sense (except perhaps
trivially) in the complex numbers.

On the other hand, in the Guassian integers (ring of numbers of the
form a + b*i where a and b are integers), there /are/ "prime numbers"
and "composite numbers". But there are also "units", which are
/neither/ prime nor composite; and i is neither prime nor composite in
the Guassian integers.

Primeness and compositeness are not properties of the /symbol/ "i".
They are properties of the "number" i /in a particular context/. Change
the context, and the meaning of these properties may change, and even
become meaningless /in that context/.

Cheers - 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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