Harristotelian Logic

Some recent threads simply explaining yet another way to see the coverage problem of the ring of algebraic integers 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

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, let's choose x = sqrt(5) - 2, considering only the positive solution, then in complex numbers, I have

S = 1/(1 - sqrt(5))

which is a complex number, and it's also an algebraic number, but it's NOT an algebraic integer.

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!

Yet you can approach it out to infinity, so you have an asymptotic approach to that value.

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!!!

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!

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.

But now you have a problem with

S = 1 + x*S

as it's just not necessarily true in the ring of algebraic integers.

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.

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.

[A repy to someone who noticed that James had previously asserted that if all the terms in a series were in a ring the sum must also be in the ring.]

Well that's wrong, so yes, I had to, following what is mathematically correct.

I make mistakes, but I care about what is correct.

So I can admit when I'm wrong, as mathematics is beautiful in that what's right is absolutely right.

And people fighting for their own delusions of worth do not matter to that correctness.

After all, these arguments will one day be gone, all of you will be dead and your children, and even your children's children will be dead, but the correct mathematical arguments will still be correct.

The mathematician looks beyond the moment to any point further when all the arguing is meaningless, which is why to some mathematicians are unearthly or almost mystical.

Only true mathematicians care nothing for social crap or the accolades of the moment as they quest for absolute knowledge knowing that even God cannot change it.

And in that way, the mathematician stands in the presence of the divine.

While lesser beings stoop at the feet of social needs, begging for approval, priding themselves on social acceptance, and when they are gone—there is nothing left.