Last updated Friday 1st of March 2024

The importance of reference with regards to determining whether something is a member of itself or not

Take L to be the list of all lists.

If a list lists itself, it is a member of itself. If it doesn’t, it is not a member of itself. L is only a member of itself in L due to the following: Consider the list of all lists that list themselves. L can be both a member of L and a member of that list. However, in that list, L is not a member of itself/L (precisely because it is a member of that list). Thus, in list L, L is a member of itself, whilst in that list (or any other list other than L), L is not a member of itself.

Call any set that is not a member of itself a -V. Call any set that is not the set of all sets a V’. Call any set that’s simply a set a V (the V of all Vs = the set of all sets). 

Is the V of all -Vs a member of itself? As I will prove further on, the only V that can contain all -Vs, is the V of all Vs, and it is a member of itself. However, mainstream set theorists would probably object to this and say “when we say the set of all sets that are not members of themselves, we mean to say a set that consists of all sets that are not members of themselves, and no other sets“. What such set theorists want, is contradictory (and they acknowledge this). You cannot have a set of all sets that are not members of themselves that is itself not a member of itself. In other words, you cannot have a -V as the V of all -Vs. The -V of all -Vs, is not the same as the V of all -Vs. The former is necessarily contradictory, whereas the latter is not necessarily contradictory.

In conclusion, no V, or V’, or -V, can encompass all -Vs and nothing more. But one V can encompass all -Vs and something more. The V of all Vs encompasses all -Vs as well as itself.

Do two Vs encompass all V’s? 

Recall that L is only a member of itself in L whilst all of its other members are members of it and not themselves (in L they are members of L, whilst in their own lists (if they have one) and only in their own lists, they are members of themselves). Similarly, in the V of all Vs, only the V of all Vs is a member of itself.

The only V that is not a V’, is the V of all Vs. Thus, only two Vs encompass all V’s. One V (which is a V’) encompasses all V’s and nothing more. The other V (which is not a V’) encompasses all V’s and something more. The latter is the set of all sets (the V of all Vs), the former is the V’ of all V’s (the not-the-set-of-all-sets set of all not-the-set-of-all-sets sets). Note that there is no V beyond all Vs and that only one V encompasses all Vs and nothing more (the V of all Vs).

Whilst there can be two Vs that encompass all -V’s (-V’ = any V’ that is not a member of itself), there can only be one V’ that encompasses all -V’s. But no V encompasses all -V’s and nothing more.

Proof that only the V of all Vs can encompass all -Vs

-V is meaningful because of V (just as -V’ is meaningful because of V’). It is in the definition of -V that it is a V that is not a member of itself, precisely because it is a not-the-set-of-all-sets set (a V’) that is a member of the V of all Vs (which makes it a -V as well as a V’). The V’ of all V’s is not a -V’, but it is a -V. This (coupled with the last sentence of two paragraphs ago) is proof of how only the V of all Vs can encompass absolutely all -Vs.

“Limitations” in forming new sets

How about forming a new set by taking all sets that are members of themselves and putting them in a new set (essentially forming a set that only contains all sets that are members of themselves)? How about forming a new set by taking all sets that are not members of themselves and putting them in a new set (essentially forming a set that only contains all sets that are not members of themselves). If such was the approach/standard, not only is the set of all sets that are not members of themselves a set that only contains all sets that are not members of themselves impossible, but so is the set of all sets that are members of themselves a set that only contains all sets that are members of themselves. This is down to the semantical implications of the word ‘all‘. Suppose we were to take all sets that are members of themselves and put them all into a new set. Is this new set a member of itself? How could it be a member of itself when prior to its formation, all sets that were members of themselves were taken and put into it? Either we did not put all sets that were members of themselves into a new set, or, we put all sets that were members of themselves into a new set. If the latter, then this new set was not a member of itself. If the former, then we did not put all sets that were members of themselves into a new set.

If we were to form a new set, we would have to add it to the V of all Vs. This cannot be done because the V of all Vs encompasses absolutely all sets (it is not just the set of all sets that exists here and now. It is the set of all sets). Trying to add to it would be like trying to go beyond infinity. Of course, the unlimited is not limited just because it cannot be added to. It is because it is unlimited that it cannot be added to.

Conclusion

If we are to be absolute with our standards or reference in relation to sets, then only the V of all Vs is a member of itself. If we lowered our standards (for example, we lower our standards from Vs to V’s), then other Vs can be interpreted as members of themselves (the V’ of all V’s was one such example), but I reiterate, the V’ of all V’s is only a member of itself in the V’ of all V’s, and that it can only be a member of itself as a V’, and that all V’s are Vs (just as all even numbers are numbers). By definition, no V’ that is a -V’, can ever be a member of itself as a V’. Similarly (but more absolutely), no V other than the V of all Vs can be a member of itself as a V (all other Vs are –Vs). The V of all Vs is the pure/true/universal/absolute/complete V because it is not a V‘, or a –V‘, or a V” or a –V (all of which are –Vs). It is a V that is in no way limited in its V-ness.

The universal set

Call absolutely any thing (number, shape, tree, human, dream, colour, set, item of imagination) an ‘existent’. Call the set of all existents ‘Existence’. Note that I am not referring to how real something is/exists, just that it is an existent (a member of Existence). Numbers are numbers (which is the same as saying numbers exist as numbers in Existence). The alternative is to say numbers don’t exist in Existence, or that there is no such thing as numbers (like round squares or other contradictory things. Contradictory sentences, beliefs, and people exist, but round-squares do not exist). Since all existents are members of Existence, only Existence is a member of Itself as an Existent.

By definition, Existence has no beginning and no end. Rejecting this yields contradictions:

It is hypothetically possible to have more than one galaxy or planet , but it is impossible to have more than one “Existence”. By “Existence” with a capital E, I mean that which encompasses absolutely all existents. Without Existence, no Existent would encompass or unify all things/existents into one Existence. This would mean that it is possible for one set of existents to be in existent A, and another set of existents to be in existent B, such that no Existent encompasses A and B. Since no Existent encompasses A and B, this means that non-Existence separates A from B. For non-Existence to separate A from B, it would have to Exist. It is contradictory/absurd (semantically inconsistent) to say non-Existence separates A from B because non-Existence does not Exist for it to do this. Hence the necessary existence of Existence. Semantics exist in Existence, as do imaginary unicorns (I imagined one just now). How real something is in Existence, is another matter.

Existence is a meaning; is It a member of the set of all meanings? Existence is the set of all meanings because there is no other thing, existent, set, or meaning that existentially encompasses all meanings. The set of all ducks is not some existing animal or shape. The set of all ducks is Existence Itself (which is an Existing meaning/set/existent). In other words, all ducks (imaginary, dream, or otherwise) exist in Existence. An imaginary duck exists as an imaginary duck. Dreams and imaginary ducks may not exist/be as real as us, but they are not non-existents like “round-triangles”. Since only Existence is Infinite (the capital I is to emphasise that I am referring to actual infinity as opposed to what some would call potential infinity), Existence is the set of all cardinalities.

Only one Infinity

If I count 1, 2, 3, 4 ad infinitum, will I reach Infinity? One cannot count to Infinity, and even if something like a number sequence goes on forever, it will not reach Infinity. To call {1,2,3,4,…} an Infinite set is to imply that {1,2,3,4,…} consists of an Infinite number of numbers. No doubt, even if 1, 2, 3, 4 goes on forever, an Infinite number of numbers will never be reached. So the question that must be asked now is whether there is any meaningful difference between 1, 2, 3, 4 ad infinitum and {1,2,3,4,…}.One might argue that the latter encompasses imagining that the count to Infinity is complete, but one cannot imagine such a thing. I don’t believe there is any meaningful difference between 1, 2 ,3 4 ad infinitum and {1,2,3,4…}.

Have any successfully counted to Infinity using x and then successfully counted to infinity using y and then compared the two countings to be able to meaningfully say something like “this Infinity is bigger than that Infinity”? There isn’t one Infinity that’s exclusively encompassing of x and another that’s exclusively encompassing of y for any to be able to “compare the sizes of two different Infinities”. Infinity is the true universal set. It is the set of all cardinalities, houses, beings, possibilities, and so on.

Infinity is meaningfully “reached” when one talks about Existence or the set of all existents in a truly absolute sense. In other words, the only reason something like a sequence of numbers can go on forever, is because of Infinity. Two different things can go on forever at different speeds, but this does not mean that one will go farther than the other when both are set to go on forever. It may look that way if you were to try and “map the distance covered by one to the other”, but neither will ever cover an Infinite amount of distance for one to be able to conclude something like “this Infinite distance covered is greater than that Infinite distance covered”. Of course, this is not the same as saying something like “this amount of distance covered in Infinity is greater than that amount of distance covered in Infinity”.