Egg Theory

Egg Theory (sometimes spelled Eg Theory to avoid confusion with the common food item) is a Dwaian mathematical theory used in a wide range of applications. It has been used by many theorist as an axiomatic foundation for the entirety of mathematics, a notable example being Smerg's Egg Theory with the Axiom of Bois (SETAB). Its fundamental object is the egg, which acts as both a collection and a transformation, making Egg Theory like a mix between Set Theory and Category Theory.

Egg

Egg Theory defines one fundamental type of object, the egg. An egg is defined by the following 6 axioms:

E1 (Content): An egg may contain other eggs, or be empty. (If E and B are eggs and B contains E, then E is called an egument of B, and B is called a box or boss egg of E. "B contains E" is written as BeE.) An egg may also contain itself.

E2 (Containment): Every egg A has a box egg (denoted b(A)), except for the universal egg, denoted U. No egg has more than one box.

E3 (Gestation): Every egg A has a source or pre-egg (denoted s(A)) and a destination or post-egg (denoted d(A)), both of which are eggs. For every egument E of s(A), A gestates E into its A-chicken, A(E), which is an egument of d(A).

E4 (Composition): For any two eggs A and B, if d(A) = s(B), then there exists a composition of A and B (denoted AoB) such that for every egument E of s(A), we have (AoB)(E) = A(B(E)). Additionally, we define s(AoB)=s(A), d(AoB)=d(B) and b(AoB)=b(A)ob(B).

E5 (Identity): For every egg A, there exists the identity gestation I_A, with s(I_A)=A, d(I_A)=A, and for every egg E in A, I_A(E)=E.

E6 (Equality): If two eggs have the same eguments, box, pre-egg and post-egg, and perform the same gestation, they are equal.

One of the elementary consequences of the egg axioms are that the elements of AoB, if it exists, are of the form CoD, where b(C)=A and b(D)=B.

Mentions

Egg Theory is mentioned by Smerg in the Dwaia anime series, S1E14, where Blombo and the gang are discussing the source of the seemingly endless knowledge of the Cube of Answers. Smerg suggests that a particular formulation of Egg Theory could allow one to reduce all questions about the universe to a polynomial form, given enough computational power, using something called "Pseudo-Infinite-Arity Eigenstochastic Klein-Cardinal Left-Matrices over Skew-Cyclic Post-Octonion Meta-Forest Half-Antimanifolds[sic]". The others dismissed his idea because it made too much sense.