A formal theory of memes, part 1

*A formal theory of memes, part 1*

We start by stating that our primitive notions are those of meme and description.
We axiomatize the theory of memes as follows:

Axiom 1 (equivalence of memes): A meme can be characterized by its full description: if two memes allow equivalent descriptions, then said memes are the same. Note that two different memes could share a description as long as they differ in a second description.

Axiom 2 (principle of no self-reference): From every meme, you can extract an element of it such that if this element looks like a meme, it does not share any description with the original meme.

Axiom 3 (principle of meme propagation): It is possible to create arbitrary memes by taking variations of descriptions of a fixed meme.

Axiom 4 (principle of stealing components): a valid meme can be formed if you steal a describable idea from two memes and put them together in your meme.

Axiom 5 (principle of stitching): a valid meme can be formed if you put together the descriptions of two memes.

Axiom 6 (principle of changing the character): a description of a meme that leaves a subject to vary can be evaluated to any particular subject insurance and form a good meme.

Axiom 7 (existence of an infinite meme): there exists a meme that allows infinitely many descriptions.

Axiom 8 (principle of generalization of memes): with any given meme, you can form a valid meme by considering all the possible combinations of descriptions of memes at the same time as a meme.

Axiom 9 (indescribable meme): a meme with no descriptions exists.

Axiom 10 (ordering of descriptions): the descriptions of memes can be sorted in a way where every collection of descriptions of the meme has the worst description.

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Note that Axiom 2 implies that no meme admits itself as a description. Note also that hypothesis 9 is not contradictory with our theory since a description of a meme is a primitive notion.

A meme can rarely be described with finitely many descriptions, so we ought to be careful with our examples.
1. A big chungus meme is not equivalent to a cheems meme because they have two different particular descriptions.
2. If you take doge and cheems, both members of different memes, you can create a meme with both.
3. It is almost impossible to construct an explicit order of descriptions of a meme.

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Proposition 1: there is only one indescribable meme.
Proof: suppose M and N were different memes, and M is indescribable. Then by axiom 1, they must differ in at least one description. Since M has no descriptions, N must have at least one description; hence it cannot be indescribable.

Proposition 2: the meme of all combinations of descriptions of a given meme has more descriptions than the original meme.
Proof: let m be a meme and M be the meme composed of all the combinations of descriptions of m. It is clear that it has at least as many as the original. For them to be equal, every description of m should correspond to one of M and all descriptions of M should be achieved in such way. But that correspondence would not be able to achieve all descriptions of M, because the collection of descriptions of m which are not achieved by the correspondence, is a valid description of M and, by definition cannot be achieved.

Proposition 3: There is no meme of all memes, that is, a meme that admits all descriptions
Proof: if m was such a meme, we could construct a meme M consisting of all combinations of descriptions of m (Axiom 8), and by proposition 2, it would have more descriptions than the meme of all memes, but that's not possible.
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Proposition 4: Let M be a meme such that two descriptions D and D′ of M are the only two descriptions. For all m we have memes composed of D and D′ (i.e., W=memes composed of  D) and memes composed of all descriptions except D (i.e., U=memes composed of D′). Then M= W∪U.
Proof: by definition of M, W is a meme composed of D, and U is a meme composed of D′. If we add all descriptions, not in these memes, we get a meme composed of all the descriptions, i.e., the original meme.

Proposition 5: The memes composed of the only description of a meme are the same.
Proof: since they are composed of only one description, they differ in no description; hence they are the same meme.

Proposition 6: The meme composed of all descriptions of another meme is not equivalent to it.
Proof: suppose there were two memes, M1 and M2, such that M1 was composed of all descriptions of M2 and vice versa. Then, by axiom 1, there would exist at least one description of D such that D (M1) = D (M2) = D and D (M2) = D (M1) = D, but this means that M1 = M2 and thus axiom two would imply that these memes admit their description as an element, which is not possible.

Proposition 7: any equivalent memes are composed of the same descriptions. Proof: since equivalent memes have equivalent descriptions, there exists at least one description such that every description of one meme is equivalent to every description of another meme; hence if we compose them all in a single meme, we get the original meme.
Proof: suppose there was a mapping φ such that φ( A) was an equivalence relation on A so that A ⊆ [φ(A)]. Suppose also that φ(A) was an equivalence class whose size was 1 (i.e., A=φ(A)). Then φ(A)=φ(A)=φ(A)=φ(A)=φ(A)=φ(A)=…=φ(A) with ∞<|φ(A)|<∞ members in A=∪[φ(A)] which contradicts our assumption on φ; hence there cannot exist such a mapping φ. This implies that if two objects are equivalent, their size is infinite (i.e., their number admits no upper bound).
Proof: Let f be the function from A to A where f([x])=[x] for all [x] in A; then f is an injection from A to A, and hence its image admits an upper bound "a." If "a" were finite then f([x])∈f([y]) for any [x],[y] in A would imply "a"≤"b," where [x],[y] in A; hence "a" would be infinite since it admits no upper bound, but that contradicts our assumption on f so "a" must be infinite as well (the image has no upper bound). Hence it follows that for every "a", there exists [x], for which f([x])=f([y]), i.e., every object has an infinite number of equivalence classes (i.e., they are not finite).
If some object has no equivalence classes, then it is different from any other object, and hence different objects have different sizes; but this contradicts our assumption on f, and hence this case cannot happen, i.e., all objects have equivalence classes (which means they have infinite size). Now suppose φ(A) admits infinite equivalence classes. It follows that φ(A)=A, so A⊆φ(A), and hence φ is a homomorphism from A to A whose image is not A; but if an image is not the whole set, it cannot be a subgroup; thus there can be no injective homomorphisms from A to A; hence φ can be none and we conclude that memes are different from the original meme (in particular, they admit descriptions).

Proposition 8: It is impossible to construct equivalence between memes in the form of an isomorphism between their structures. Proof: suppose A and B were two memes, and there was a mapping f: A→B such that f: A→B was an isomorphism (between their structure). But since A and B are sets of descriptions, this implies that there must exist two equivalent objects in A so that their equivalence implies the existence of the same objects in the same number in B;  but that's not possible since an equivalence does not necessarily preserve the size.

Proposition 9: no meme contains all descriptions.
Proof: suppose the meme contained all descriptions. Then, by axiom 1, it would contain no descriptions. Then, if it contained descriptions, we could build a meme consisting of all descriptions of it (i.e., the original meme).

Proposition 10: the description of a meme cannot contain all descriptions of another meme.
Proof: if so, we could construct a meme by combining all descriptions of two memes, A and B, and this would imply that all the descriptions of memes would be contained in it, which is impossible.

There is a meme that admits only one description.
Proof: suppose the meme m had two or more descriptions, then the meme m∪all memes would have more than one description.