Hi,

I stumbled of the Cuil Theory and was mildly interested in it. However, as a undergraduate maths student, one thing is bugging me: The "levels of abstractness" is not well defined. Instead I have thought about a alternative definition:

Firstly, I will tell you something about cuntinuverse theory: This is an expansion of the Multiverse theory proposed by quantum physicists in the early 1900s. It does not assume a countable (or even finite) number of universes, but instead a Infinite-dimensional universe, in which every universe, EVERYTHING exists. The Idea of it is: 3D space is where we live. Adding a 4th dimension, we have a sequence of 3D spaces, and we could call the 4th coordinate "Time". The 5th dimension combines multiple timelines, in 1 universe, with the same starting point, with the same laws of nature. Next, the 6th dimension modifies the starting point (Usually the "big bang"). In this dimension, you find 5 dimensional universes that started of with a different, or no big bang at all. This can be continued on and on and on, but I dont want to bother you. But be told that a 8 dimensional universe allows you to find a universe in which you are actually a radio wave that has a terrible, terrible headache while buzzing around a hamburger that is made out of milk.

Here we get really close to Cuil theory and we can define (integer) Cuils:

A **event** A is a subset of the infinite dimensional universe U.

We define the **embedding space** of A $\text{em}(A)$ as the lowest-dimensional submanifold S of U that contains A and the **dimension** $\dim(A)$ of A as the dimension of S. (For things we can imagine, this is just the same dimension as we would call it. A (infinitely thin) piece of paper would have dimension 2, Your dog has dimension 3 and your dog wagging its tail dimension 4)

For 2 events A,B we can now define the Cuility of B towards A (measured in Cuils) as

(1)On top of that, if it is clear which A we use, we can write $Cuil(B):=Cuil(A,B)$. In most cases, A will be our 4-dimensional spacetime (where I ask you for a hamburger).

This almost holds with the layers of abstraction in general Cuil theory. If I ask you for a hamburger, that is a 4 dimensional event. If you hand me a hamburger and I turn into a dog wagging its tail while balancing on a 1-dimensional radio wave with a headache, that would be Cuil 4: $\dim A =4$, $\dim B =8$, $\dim (A\cup B)-\dim A=8-4=4$.

Secondly, lets look at alternative definitions for the dimension for an event: For example, using hausdorff dimension (or similar), a Menger sponge has dimension $\frac{\log 20}{\log 3}\approx 2.7268$. If you hand me a Hamburger in the shape of a Menger sponge, that would be either dimension 4 or 7 (depending on if its actually possible to build a true menger sponge in this universe), however, a radio wave inside a Menger sponge would have dimension 5.7268 or 8.7268, and, of course, terrible, terrible migrane.

EDIT: Continuation.

We can define Operators $+$, $-$, $\cdot$ and $/$ by

$Cuil(A,B)+Cuil(A,C):=Cuil(A,B\cup C)$

$Cuil(A,B)-Cuil(A,C):=Cuil(A,B\backslash C)$

$Cuil(A,B)\cdot Cuil(A,C):=Cuil(A,B\times C)$

$Cuil(A,B)/Cuil(A,C):=Cuil(A,B/ C)$

This highly depends on the actual events.

Examples:

A = You ask me for a hamburger. (dim=4)

B = I give you a hamburger. The hamburgers eye twitches involuntarily (dim=6, 2‽)

C = A hamburger goes for a stroll in time. I give you a racoon. (3‽)

B+C = I give you a hamburger. The hamburgers eye twitches involuntarily and a hamburger goes for a stroll in time. Then, I give you a racoon (dim=7, 3‽)

B-C = I give you nothing. (dim=4, 0‽)

B*C = I give you a hamburger. Because the hamburger is going on a stroll in time, we revolve around history. The racoon gives me the hamburger. Time twitches. I give you a stroll in time while my racoontwitches involuntarily. We see every possible happening and every timeline get eaten by you. The racoons eye twitches. I give you a hamburger. Nothingness commences as the hamburger reaches its destination. I eat time, my watch is gone. We take a stroll in a hamburger. The racoon runs to the twitching eye. It shakes. I give you a hamburger. We come to a halt. I give you a hamburger. We twitch in all directions, at the same time and after one another. I give you a twitching hamburger. The twitching hamburger gives me a racoon. I give you a hamburger. The hamburgers eye twitches on a stroll in time. I give you a hamburger. The hamburger strolls along a path in time that follows the contours of the racoon. *(… (infinite other events follow here) …)* (dim=6*3=18, 18-4‽=14‽) *In fact, every possible event that can be created by the two starting ones, happens*

B/C = The hamburger is a racoon. The hamburgers/racoons eye twitches involuntarily. (dim=5, 1‽)

A = Inside or a menger sponge, I feel pain in my head, because I am a radio wave (4.7268‽)

B = The hamburger sits on a bench, sleeping. (1‽)

C = I carefully open a book. I look at the stars. They are grains of sesame on the hamburger. The universe is contained in the sponge.(5.7268‽)

Excercise: Calculate these results: B+C, B*C, B-C, B/C. Good luck with not getting your brain exploded.

cu, CBenni