The Opticon



Whereas, previously, I tried to make sense of the surreal numbers, after a little more study, I've come to a more reasonable understanding of the subject. My knowledge on it is not yet complete, though.

Let's imagine the Real Line of numbers. It extends from zero in some direction, and in the direction opposite to it. Around the origin, there's what some mathematicians like to rightly call the Infinitesimal Monad.

Since the Opticon is a scale-construct, it follows the Law of Continuity, by Leibniz: "the rules of the finite are found to succeed in the infinite." Therefore, there's an operational limit over the infinitesimal quantities. This means infinitesimal quantities are incomparable to real quantities up to exponentiation. Similarly, there's an operational limit to the Reals. Namely, infinity, ω_1, the first uncountable ordinal, the limit ordinal of the Reals.

Now, in the Reals, the first limit ordinal greater than all infinitesimal quantities is, by necessity, the number 1, the scale unit element. So, if we give the set of Reals a surname, let us call it R_1. Therefore, let us call the infinitesimal Monad R_0, since an infinitesimal number is still a real number. Now, there are still Real numbers beyond the ω_1. These Reals, we call R_2, and its infinitesimal Monad is the infinite union of all R_n, for -ω < n < 2. Therefore, for every R_k, its monad is the union from n = k-1 to -ω of R_n.

There is a function f (x) = (ω_1)^x that represents the operational limit for some scale. For R_1, the operational limit of the scale unit, f (1) = (ω_1)^ 1 = ω_1.

Notice that, being f (0) = 1, f (1) = f (f (0)). Then For every R_k its operational limit is given by the recursive composition of f (0) k times. We denote this as [k]f (0). Furthermore, the inverse function of f (x) is f^(-1) (x) = x^(1/ω_1). So, the unit element is [k-1]f^(-1) (ω_1), and the operational limit of its Infinitesimal Monad is [k]f^(-1) (ω_1).

This concludes the basic definition of the Opticon.

Comments

Post a Comment

Popular posts from this blog

Crítica a la Industrialización en el siglo 21

A medida en que la industrialización abarató los costos de producción, pudimos observar cómo, en general, la calidad de vida mejoró primordialmente entre los países más desarrollados, dado al servicio de mano de obra barata en países donde la pobreza es extremadamente abundante. Recientemente, he recibido noticia, que puede ser falsa, de que ante los impuestos de arbitrios de intercambio en el mercado internacional, varias fábricas en países como China, encargadas de la manufactura de objetos que luego serían vendidos bajo alguna marca de lujo se vieron obligados a vender los mismos objetos que producían, pero simplemente, sin la marca de lujo cuando las marcas de lujo comprendieron que no les sería rentable seguir comprando obra barata en China, por ejemplo, para luego venderlas al pueblo norteamericano. Por un lado, es impresionante la mentira que la industrialización presenta hoy día, en que el valor del objeto de lujo es una ilusión de mercadotecnia y publicidad. Por el otro, es un...
Read more

Last Night in Cairo

Last Night in Cairo   Outline © 2023 G. M. Dávila Andino 1. Dr. Seawell has just arrived in Cairo. He says goodbye to his photographer friend. They were on an expedition through North Africa. The Photographer’s going to Gaza to cover the war. 2. He takes some time to walk around the city, just to get a feel for it. Maybe he buys some food from the local cuisine at a small venue. A snack, or a coffee. 3. Veronica's on vacation with her best friend, trying the local cuisine in Cairo, road trip to Giza, buying textiles for her custom clothes. This year has been good for her. 4. At the café, there’s a missed meet between Alexander and Veronica.  5. He buys a prepaid phone and a phone card, maybe, and a computer, he leases a car, parks it, checks into the Marriot Cairo Hotel, checks his emails. there’s one from Anastasia Radwizill. 6. He calls her, leaves a voice message, looks at his Omega watch and passes out. It’s just late and he’s tired. 7. Dr. Seawell wakes up t...
Read more

Joanne Bergson 2.01

The day strikes its first notes of blue as Joanne brews a few cups of her favorite coffee brand. With the click of a button the news start playing on the TV. Terrorists have bombed the Channel Tunnel that runs between London and France, killing thousands and causing an exorbitant amount of money in damages. The world is outraged and the only clue to who may have done it is a video left on a popular streaming website, from an anonymous account. In the video, an artificially generated voice over a completely black screen identifies itself as an anarchist military group called Enkidu and proudly announces they are responsible of the attack on the Channel Tunnel, claiming that civilization is the source of all human problems. They promise that this will not be the end—we should all be afraid. Very afraid. Joanne Bergson shrugs off the news. She’s got more important things to think about. Like what she’s going to wear tonight for her first webcam show, and how well would her friend Genevi...
Read more