A new definition of subset

 1. I don't think it is necessary to have two words to indicate parts of a set, namely, subset and proper subset.

2. Let's drop "proper subset" and define "subset" to be the current definition of "proper subset". This is what mathematicians do in practice anyway. They use "subset" when they mean "proper subset". 

The above screenshot is a Mathematica function that lists all "subsets" of a set. But except the last one (equality) they are all "proper subsets". This means that mathematicians do not respect their own definitions of "subset" and "proper subset".

A new definition of subset   

1. Given two sets A and B, we say that B is a subset of A if every element of B is an element of A but A has at least one more element.

2. In other words, given that $\mathbf{A} \neq \mathbf{B}$, $\mathbf{B}$ is a subset of $\mathbf{A}$ if all the elements of $\mathbf{B}$ are elements of $\mathbf{A}$ and $\mathbf{A}$ has at least one more element.

3. This definition covers all possible situations a set can be partitioned except when $\mathbf{A}=\mathbf{B}$.

4. If $\mathbf{A}=\mathbf{B}$ it is silly to say that $\mathbf{A}$ includes $\mathbf{B}$ or that $\mathbf{B}$ includes $\mathbf{A}$. Since they are equal neither is a part of the other. No Cantorian rhetorical sophistry can change this fact. It's funny that mathematicians accept easily and never question rhetorical sophistry if it comes from a dead mathematician with fame and authority.

5. There are various ways to draw equal sets as subsets with Venn diagrams.

6. If $\mathbf{A}$ and $\mathbf{B}$ are equal we call them equal. There is no reason to define the word "subset" as a synonym to the word "equal". There is no subset relation between sets that are equal.

7. The "subset" and "proper subset" jargon is invented in order to claim that equal sets are subsets of each other, that is, one is a part of the other. But if $\mathbf{A}=\mathbf{B}$ neither is a part of the other.

8. Mathematicians corrupted the words "whole" and "part" by denying Euclid's Common Notion 5: The whole is greater than the part.

9. In this case we have two wholes: $\mathbf{A}=\{1,2,3\}$ and $\mathbf{B}=\{1,2,3\}$

10. $\mathbf{A}=\mathbf{A}$ must be different than $\mathbf{A}=\mathbf{B}$.

11. To say $\mathbf{A}=\mathbf{A}$ you need to clone $\mathbf{A}$ and create a new set. We must name this new set with a name other than $\mathbf{A}$, e.g., $\mathbf{A'}$ and then we say $\mathbf{A}=\mathbf{A'}$

12. $\mathbf{A}=\mathbf{A}$ doesn't make sense because a whole is not a part of itself.

13. No one is saying that equality implies whole/part relationship but when mathematicians say that every set is a subset of itself they are implying that the whole is part of itself.

14. To say that "a set is a subset of itself" is rhetorical sophistry and play on words "whole" and "part".

15. I respect Euclid's Common Notion 5 and I don't see any reason to deny that the whole is greater than the part. This means simply that the big is greater than the small. Only paradox loving set theoristas deny this Common Notion to save their beloved set theory.

16. If we accept and respect Euclid's Common Notion 5 that the whole is greater than the part, a set cannot be a subset of itself because the whole (the set) cannot be a part (subset) of itself. Equals will not contain one another except in the absurd world of the set theory.

17. The part cannot be equal to the whole. To say that the whole = the part is sophistry because the definitions of "whole" and "part" are being redefined on the fly and on the sly.

18. What justifications do mathematicians have to deny Euclid's Common Notion 5?