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[jstevh@xxxxxxx]
> Some recent threads simply explaining yet another way to see the
> coverage problem of the ring of algebraic integers
What does "coverage problem" mean, precisely? Which specific result of
Dedekind's, or of ideal theory in general, do you believe is in error? And
why is it that you never answer these simple questions?
For the rest, I must say this is one of the clearest posts I've seen from
you in years. I don't know what you think the /point/ of it is, but at
least most of it was intelligible.
> have floundered on the issue of convergence and what it means with
> rings like the ring of algebraic integers, where I introduce the
> concept of an exclusionary ring.
>
> The concepts are simple, luckily, as I can mostly use something basic:
>
> S = 1 + x + x^2 + x^3 +...
>
> where the issue is convergence, so that if S converges in whatever ring
> you're in--notice none given yet--you can go to
Except you have no idea what mathematicians mean when they talk about
convergence. I understand that you /think/ you do, but you're mistaken
about that (which is obvious to everyone who knows something about the
topic).
But let's skip all that, and just assume you have some fuzzy notion of
"differences of real numbers getting closer to 0" in mind.
> S = 1 + x*S
>
> and solve to get
>
> S = 1/(1-x)
>
> where to get convergence in the ring of complex numbers, which of
> course is also the field of complex numbers,
Stick to the reals. You don't need complex numbers for this example.
> let's choose x = sqrt(5) - 2, considering only the positive solution,
? sqrt(5)-2 ~= 0.236 /is/ positive. That's like saying "chose x = 3,
considering only the positive solution". The longer half of it sounds like
gibberish.
> then in complex numbers, I have
>
> S = 1/(1 - sqrt(5))
No, you have S = 1/(1-x) = 1/(1-(sqrt(5)-2)) = 1/(3-sqrt(5)). You should be
able to see instantly that the expression you came up with for S is
negative -- which doesn't work all that well since you also want to believe
it's an infinite sum of strictly positive terms ;-)
> which is a complex number, and it's also an algebraic number, but it's
> NOT an algebraic integer.
I didn't check those claims, but am happy to assume they're true (in case
they're not, it's certainly possible to find specific values where they are
true).
> But if I go back to
>
> S = 1 + x + x^2 + x^3 +...
>
> and plug in x=sqrt(5) - 2, I'll have
>
> S = 1 + (sqrt(5) - 2) + (sqrt(5) - 2)^2 + (sqrt(5) - 2)^3 +...
>
> and if I stop at some point like just look at
>
> 1 + (sqrt(5) - 2) + (sqrt(5) - 2)^2
>
> I have an algebraic integer, and in fact, I can have an arbitrarily
> large number of terms added together, and stop--ever closer to
> 1/(1-sqrt(5)) and STILL have an algebraic integer.
>
> BUT in the ring of algebraic integers, you cannot reach 1/(1-sqrt(5))
> because it is NOT an algebraic integer!
Sure. And ...?
> Yet you can approach it out to infinity, so you have an asymptotic
> approach to that value.
OK. And ...?
> A corollary to this is 1/x where you can let x go out to infinity and
> that approaches 0 but never reaches it, or you can let x approach 0,
> but never reach it, but the difference with the infinite sum example is
> that the asymptotic nature is created by the exclusionary nature of the
> ring of algebraici integers!!!
What of it? This is common as mud, in /all kinds/ of rings. Nothing in the
definition of a ring or of a field requires that a ring or field be closed
under infinite sums, and regardless of what notion of convergence is applied
to infinite sums.
For example, under the only notion of convergence you vaguely know about,
the infinite sum 1+1+1+... in the ring of integers isn't even defined.
Every finite prefix of the infinite sum 1 + 1/1! + 1/2! + 1/3! + ... in the
field of rationals is a rational, and it converges too, but to the
irrational (more, transcendental -- not even algebraic) real e ~= 2.71828.
Etc, etc, etc. This aren't "problems", they're just facts.
> That is, the reason you can never reach 1/(1-sqrt(5)) with the series
> is that 1/(1-sqrt(5)) is NOT the root of any monic polynomial with
> integer coefficients, which is the rule that defines algebraic integers
> and excludes that value!
What of it? The algebraic integers "exclude" uncountably many reals; so
does the ring of algebraic numbers; so does the ring of rationals; so does
the ring of integers. All of those rings are countably infinite, so they in
fact /have/ to "exclude" almost all reals.
For some reason, you seem to take personal offense at that when, and only
when, the algebraic integers are the ring in question :-(
> So the mathematics holds on a definition, as logically, if
> 1/(1-sqrt(5)) is not an algebraic integer, then there is no way to
> reach that value in the ring, so
>
> S = 1 + (sqrt(5) - 2) + (sqrt(5) - 2)^2 + (sqrt(5) - 2)^3 +...
>
> approaches it asymptotically, in the ring of algebraic integers.
What of it?
> But now you have a problem with
>
> S = 1 + x*S
>
> as it's just not necessarily true in the ring of algebraic integers.
There's no problem here. NONE. Forget infinite sums for a minute: rings
aren't necessarly closed under division, and you need division just to get
from S=1+x*S to S=1/(1-x). There's no /a priori/ reason to imagine you can
do that in /any/ ring.
It would be just as groundless to complain that "there's a problem" with S =
1 + x*S in the ring of integers, "because" given any integer x except for 0
and 1 there is no solution for S in the ring of integers.
BFD. The integers are closed under some operations but not others; ditto
the rationals; ditto the algebraic integers; ditto the algebraic numbers;
ditto the reals. If you took a course in abstract algebra, you'd learn that
a great deal is known about how to embed a structure you care about in a
larger structure that's closed under the operations you care about. For
example, the complex numbers are the so-called "algebraic closure" of the
reals, just as the algebraic numbers are the algebraic closure of the
rationals.
When a given structure S isn't closed under a given operation O, that's just
a fact about S and O, not "a problem" (although it may be a challenge to
prove closure one way or the other).
> Why not necessarily true?
>
> Because the exclusion does NOT apply if 1-x is a unit in the ring of
> algebraic integers.
>
> So if it is a unit then the series can converge within the ring, but
> otherwise the exclusionary rule--the definition of algebraic integers
> as roots of monic polynomials--prevents.
>
> Now I feel confident there are posters who will want to dispute me on
> the ability to use convergent infinite series in the ring of algebraic
> integers, but I want more than namecalling, or other childish rants,
> and I want more than someone saying they read it different in some
> number theory text.
You can use infinite sums all you like, provided you're careful to use a
sensible meaning for convergence, and spell out what that is.
Skipping the rest, because it's just another rant:
> The problem here again is that people who don't know mathematics that
> well, can write books.
>
> And other people can believe things that are wrong for quite some time,
> and LOTS of people can believe things that are wrong, so simply saying
> a lot of people have believed other things for a long time is not the
> way to reply back to me here.
>
> If any of you actually know mathematics versus being people who can
> repeat what you're told as if that's all that matters, then you can
> explain the status quo view.
>
> You can explain the mainstream beliefs of the mathematical community in
> this area without relying on insults, without simply claiming that I'm
> wrong, and without doing anything other than objectively replying.
>
> I know that can't be done, so I say that upfront to remind readers that
> a lot of people around the world are fighting the truth with my
> research to preserve their delusion of expertise, which is all about a
> lot of people getting some mathematics wrong.
Which specific result of any established mathematical theory do you believe
is in error? Note that I want "more than namecalling, or other childish
rants", I'm asking a specific question about mathematics. You claim
established theory is wrong. You say that endlessly. But you never say
/which result/ in established theory is wrong. If you can, please name a
specific theorem that's wrong in your view of the world. Or, if you can't
name a single specific result that's wrong, please explain the basis for
your claim that math professors are teaching wrong results.
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