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The axioms of set theory imply a primitive ontology in which existence is predicated on containing other things or being oneself contained. Moreover, such a description is exhaustive: when describing anything as a set, we have a complete ontology for the thing.

Second, the basic relation of all things in reality is according to their membership, to this or that set. This implies that there is nothing which exists that is not either a set or an element within a set, or both.

There is exactly one set with no members, called the empty set or ∅.

∅ ⊂ A for every A (including ∅) but it is not true that ∅ ∈ A for every A.

In English, the empty set (having zero elements) is a subset of all sets, but it isn't true that ∅ is an element of all sets. We'll see why below.

So, if we take A = {1, 2, 3} and we begin to remove elements, we can show that ∅ is a subset of A. (Sn are subsets of A).

Sx ⊆ A = {1, 2, 3}

Sy ⊂ A = {1, 2} (one element removed)

Sz ⊂ A = {1} (two elements removed)

∅ ⊂ A = {...} (all elements removed)

Note, that this does NOT mean A = {∅}. ∅ is not an element of A, but since ∅ has no elements, it is thought to 'participate' as a subset of all sets. We might say that the membership of ∅ (that is, no members) is implied by all sets. All sets are collections whose cardinality could be be zero, therefore ∅ is a subset of all of them. ___

But what is nothing? The empty set is the one having no elements. How does this relate to the number 0? Zero is not nothing. Zero denotes cardinality. Zero would be the cardinality of the empty set, as in the number of elements it has. The empty set has some correspondence to zero in Boolean logic for computer science, but I'm not terribly interested to get into a discussion about zero: just let it rest that mathematically, zero is not nothing.

What I'm interested in is the metaphysics implied by the empty set. Most mathematicians probably have the equivalent to the empty set of reasons for finding this discussion useful (dork joke).

Someone could say that the entire inquiry is trivial: Nothing can be said about nothing. It's tautological. To speak with regard to it is just to say something, akin to {...}. By denoting the empty set this way, haven't I already missed my target? I've said something about that which has no somethings.

This is what I find metaphysically interesting (of course, I'm making all kinds of assumptions as to the limits of human reasoning and intellect). To anyone besides a metaphysician, any further discussion beyond our ability to represent the situation symbolically will seem childish.

___

It seems that there is no proper way to speak of nothing simpliciter.

In fact, whatever thought is, seems to preclude us from access to nothing by its very existence. Basic intentionality (with its aboutness) necessarily creates information, and nothingness cannot communicate information.

What ∅ signifies to me is that in probing what nothing is, the mind can only go so far as a state of affairs in which there is nothing. When we say there is nothing in the fridge, the logical extension in the statement at least picks out the refrigerator having the state of affairs in which there is no food item inside - here, even 'nothing' has an extension: food. You could say that 'nothing in the fridge' is describing the empty food set. So we don't truly mean nothing...instead we mean 0 in the presence of something: the place of food. If the fridge were a set of all its elements, F, the situation of 'nothing in the fridge' is F = {...}. This is NOT nothing.

A great deal of metaphysical debate concerning the origin of the universe involves reference to nothing, as in the possibility of the nothingness that preceded the existence of the universe. But what is it that we refer to when we speak about this possibility?

Such statements always strike me as describing any primordial pre-existence as ∅. We think of nothing and cannot do so without the brackets {...}, as in, there at least existed the state of affairs in which something could exist. We imagine vast blackness, and still, it is necessarily a thing which could contain elements. So questions about origins of the universe pivot on whether you think true nothingness is a coherent term.

  1. If nothing* is possible, then a posteriori we find ourselves in a current state of reality which must have experienced creatio ex nihilo. If there was truly nothing, this change is describable as at least the change from no state of affairs to a state of affairs, ∅, where the possibility for the inclusion of elements existed. We think creatio ex nihilo is logically impossible, so this is ruled out. If there is something today, there was never nothing.

  2. If nothing* is not possible, and something has always existed, then we must still explain ∅ according to what we know about contingent existence. To do this we'll return below to the idea of looking at sequential subsets.

Contingent existence is that which does not contain its own reason for existing, that is, it must be explained according to causes from without. An apple is contingent. You are contingent.

If the universe, U, is the finite set {1, 2, 3.....n}, it is possible by principles of causation to scientifically reduce the universe by sequential subsets as we did above, where the first subset has (n-1) elements of U. For the sake of example:

U = {f(E)1.....f(E)n} | f(E)i = { x ∈ E: u({x}) --> u({y}) for all y ∈ E }

So, the universe consists of elements which are functional states where any state can be explained by some basic function, u, which transforms some subset x to y. Granted, that's rough. But physics is pursuing an elegant theory of everything which by way of a finite set of equations hopes to do exactly this for every element of the universe - one theory which could causally reduce all of the information about the entire state of the universe.

Still, the initial state of the universe had no physical elements, and so without x and y, we're left with:

Scientific pre-universe: U = {∅, f(E)}, implying that we must at least consider the empty set an event space.

Can the universe be logically reduced to the empty set, i.e. to a state of affairs where the potential to exist is all that exists? Again this depends upon the true possibility of nothing. If nothingness could never have been the case (which we think it cannot because creatio ex nihilo is ruled out), then the reduction to the empty set is only possible where some agent brought about the first elements in the empty set.

Instead of this, particle physics might have you thinking that reality is an infinite regress of subsets of primitive particles. Others would like to appeal to the emergence of matter from natural laws, but since the LAWS of nature themselves are not actually elements, but are relations among elements, the only possible appeal would be to existence as a collection of something.

Put another way, the laws of nature could not exist prior to the set of elements whose members can relate. The universe could not have come to exist by the actions of laws relating the behaviors of matter.

Effectively, scientific reduction itself is the description of relations applying over subsets of U, to reduce them to smaller and smaller subsets, where (n-1) is thought to have greater explanatory power than (n). If the universe is a set of states, we can reduce these states to previous states, so on and so forth. We find, however, this always arrives at some subset which can neither be reduced or explained. Yet, whatever these least reducible things might be, they are contingent by definition.

According to physics, their explanation must be in laws of nature. By this mistake, the only analytical a posteriori necessity for the origin of this universe was the set { ∅, f(E) }, where we have the state of affairs for things to potentially exist, and the lawful relations which would govern events if there were any elements to be relating. But herein lies the contradiction. No relation is an element of a set, nor can a relation exist in ∅.

To get around this, physics requires an appeal to a different sort of infinite regress of elements: the Multiverse (MV).

MV = {Ux, {Uy, Uz.... ∞}}

...which says that our universe, Ux, is one element of a multiverse of infinite elements. This is how the regress to the untenable situation of U = ∅ is avoided, by introducing not an infinite regress of actual universes in causal succession, but a plenum of weighted probabilities, something more like:

MV = {P(Ux), {P(Uy), P(Uz).... ∞}}.

We just happen to find ourselves in the convenient scenario where the probability of our universe existing with just the necessary relations was = 1. Lucky us. Apparently, such an appeal to the infinitude of existence (as many possible universes), we escape both creatio ex nihilo and also the primordial state of the universe being simply ∅.

To this we ask: isn't it logically possible that the probabilities for all universes in MV are 0? Of course not, there is always a necessary element or we'd arrive at ∅. So ∅ remains a possibility for MU, contra there being a necessary universe in MV. If there is a NECESSARY universe in MV, then we have arrived at another problem, which has to do with defining the term necessary.

Someone could say, "Sure, maybe there is a necessary universe in MV and so all but (∞ - 1) universes is contingent."

Such a necessary universe could not have contingent elements. But if universes are defined as being exhaustively described by the contingent elements they contain, then the Multiverse itself could not contain any necessary elements.

What is necessary must exist separately from MV, i.e. the necessary being cannot be an element within MV. This precludes the objection where someone simply redefines what a universe is: "If there is a necessary being that exists, then MV would just come to include that thing." But if inclusion in MV means being a universal set which is itself exhausted by membership of only finite and contingent elements, nothing necessary can be included in the set MV.

As a solution to these problems, I ask: what is not an element in ANY set, but is NECESSARILY a subset proper of EVERY set?

It is ∅.

It is just the state of affairs in which the potential to exist is. One cannot avoid the intuition that the same equivalence of the foundation of the universe, ∅, has parallels with the human mind.

After all, the mind can produce no thought which refers to nothing. ∅ is the most primitive concept we can form. This would make sense if ∅ underlies a universe which is fundamentally self-simulating. Indeed, if we take mental causation seriously, then we must face the fact that we arrive at a problem similar to the Cosmological Problem itself when we consider some thoughts. Therefore, the causal explanation for certain thoughts might be thought to refer back to an analogous form of necessary being, namely ∅, or a basic plenum of creative potential, combined with the event of actualization granting Form.

We might very well think that mind cannot be explained exhaustively by the brain (though its mental powers can be locally explained, i.e. if my brain dies, my mind appears to cease to exist). But if instead we think that brains are not producing, but rather accessing the ∅, then we have a concept that looks a great deal like the Logos, an active principle which produces order from chaos.

___

The axioms of set theory imply a primitive ontology in which existence is predicated on containing other things or being oneself contained. Moreover, such a *description is exhaustive*: when describing anything as a set, we have a complete ontology for the thing. Second, the basic *relation* of all things in reality is according to their membership, to this or that set. This implies that there is nothing which exists that is not either a set or an element within a set, or both. There is exactly one set with no members, called the **empty set** or ∅. >∅ ⊂ A for every A (including ∅) but it is not true that ∅ ∈ A for every A. In English, the empty set (having zero elements) is a subset of *all* sets, but it isn't true that ∅ is an *element* of all sets. We'll see why below. So, if we take A = {1, 2, 3} and we begin to remove elements, we can show that ∅ is a subset of A. (Sn are subsets of A). >Sx ⊆ A = {1, 2, 3} >Sy ⊂ A = {1, 2} (one element removed) >Sz ⊂ A = {1} (two elements removed) >∅ ⊂ A = {...} (all elements removed) Note, that this does NOT mean A = {∅}. ∅ is not an element of A, but since ∅ has no elements, it is thought to 'participate' as a subset of all sets. We might say that the membership of ∅ (that is, no members) is *implied* by all sets. All sets are collections whose cardinality *could be* be zero, therefore ∅ is a subset of all of them. ___ But what is nothing? The empty set is the one having no elements. How does this relate to the number 0? Zero is not nothing. Zero denotes *cardinality*. Zero would be the cardinality of the empty set, as in the number of elements it has. The empty set has some correspondence to zero in Boolean logic for computer science, but I'm not terribly interested to get into a discussion about zero: just let it rest that mathematically, zero is not nothing. What I'm interested in is the metaphysics implied by the empty set. Most mathematicians probably have the equivalent to the empty set of reasons for finding this discussion useful (dork joke). Someone could say that the entire inquiry is trivial: Nothing can be said about nothing. It's tautological. To speak with regard to it is just to say *something*, akin to {...}. By denoting the empty set this way, haven't I already missed my target? I've said something about that which has no somethings. This is what I find metaphysically interesting (of course, I'm making all kinds of assumptions as to the limits of human reasoning and intellect). To anyone besides a metaphysician, any further discussion beyond our ability to represent the situation symbolically will seem childish. ___ It seems that there is no proper way to speak of nothing simpliciter. In fact, *whatever thought is*, seems to preclude us from access to nothing by its very existence. Basic intentionality (with its *aboutness*) necessarily creates information, and nothingness cannot communicate information. What ∅ signifies to me is that in probing what nothing is, the mind can only go so far as a **state of affairs** in which there is nothing. When we say there is nothing in the fridge, the logical extension in the statement at least picks out the refrigerator having the state of affairs in which there is no food item inside - here, even 'nothing' has an extension: food. You could say that 'nothing in the fridge' is describing the empty food set. So we don't truly mean *nothing*...instead we mean 0 in the *presence* of something: *the place of food*. If the fridge were a set of all its elements, F, the situation of 'nothing in the fridge' is F = {...}. This is NOT nothing. A great deal of metaphysical debate concerning the origin of the universe involves reference to nothing, as in the possibility of the nothingness that preceded the existence of the universe. But what is it that we refer to when we speak about this possibility? Such statements always strike me as describing any primordial pre-existence as ∅. We think of nothing and cannot do so without the brackets {...}, as in, there at least existed the *state of affairs* in which something could exist. We imagine vast blackness, and still, it is necessarily a thing which *could contain* elements. So questions about origins of the universe pivot on whether you think true nothingness is a coherent term. 1. If nothing* is possible, then *a posteriori* we find ourselves in a current state of reality which must have experienced creatio ex nihilo. If there was truly nothing, this change is describable as at least the change from no state of affairs *to* a *state of affairs*, ∅, where the possibility for the inclusion of elements existed. We think *creatio ex nihilo* is logically impossible, so this is ruled out. If there is something today, there was *never* nothing. 2. If nothing* is not possible, and something has always existed, then we must still explain ∅ according to what we know about *contingent existence*. To do this we'll return below to the idea of looking at sequential subsets. >Contingent existence is that which does not contain its own reason for existing, that is, it must be explained according to causes from without. An apple is contingent. You are contingent. If the universe, U, is the *finite* set {1, 2, 3.....n}, it is possible by principles of causation to scientifically reduce the universe by sequential subsets as we did above, where the first subset has (n-1) elements of U. For the sake of example: > U = {f(E)1.....f(E)n} | f(E)i = { x ∈ E: u({x}) --> u({y}) for all y ∈ E } So, the universe consists of elements which are functional states where any state can be explained by some basic function, u, which transforms some subset x to y. Granted, that's rough. But physics is pursuing an elegant theory of everything which by way of a finite set of equations hopes to do exactly this for every element of the universe - one theory which could causally reduce all of the information about the entire state of the universe. Still, the initial state of the universe had no physical elements, and so without x and y, we're left with: > Scientific pre-universe: U = {∅, f(E)}, implying that we must at least consider the empty set an event space. Can the universe be logically reduced to the empty set, i.e. to a state of affairs where the potential to exist is all that exists? Again this depends upon the true possibility of nothing. If *nothingness could never have been the case* (which we think it cannot because creatio ex nihilo is ruled out), then the reduction to the empty set is **only possible where some agent brought about the first elements in the empty set**. Instead of this, particle physics might have you thinking that reality is an infinite regress of subsets of primitive particles. Others would like to appeal to the emergence of matter from natural laws, but since the *LAWS of nature* themselves are not actually elements, but are *relations* among elements, the only possible appeal would be to existence as a *collection* of something. Put another way, the laws of nature could not exist prior to the set of elements whose members can relate. The universe could not have come to exist by the actions of laws relating the behaviors of matter. Effectively, scientific reduction itself is the description of relations applying over subsets of U, to reduce them to smaller and smaller subsets, where (n-1) is thought to have greater explanatory power than (n). If the universe is a set of states, we can reduce these states to previous states, so on and so forth. We find, however, this always arrives at some subset which can neither be reduced or explained. Yet, whatever these least reducible things might be, they are contingent by definition. According to physics, their explanation *must* be in laws of nature. By this mistake, the only analytical *a posteriori* necessity for the origin of this universe was the set { ∅, f(E) }, where we have the state of affairs for things to potentially exist, and the lawful relations which would govern events *if there were any elements to be relating*. But herein lies the contradiction. No relation is an element of a set, nor can a relation exist in ∅. To get around this, physics requires an appeal to a different sort of infinite regress of elements: the **Multiverse** (MV). > MV = {Ux, {Uy, Uz.... ∞}} ...which says that our universe, Ux, is one element of a *multiverse* of infinite elements. This is how the regress to the untenable situation of U = ∅ is avoided, by introducing not an infinite regress of *actual* universes in causal succession, but a plenum of weighted probabilities, something more like: > MV = {P(Ux), {P(Uy), P(Uz).... ∞}}. We just happen to find ourselves in the convenient scenario where the probability of our universe existing with just the necessary relations was = 1. Lucky us. Apparently, such an appeal to the infinitude of existence (as many possible universes), we escape both *creatio ex nihilo* and also the primordial state of the universe being simply ∅. To this we ask: isn't it logically possible that the probabilities for all universes in MV are 0? Of course not, there is always a necessary element or we'd arrive at ∅. So ∅ remains a possibility for MU, contra there being a necessary universe in MV. *If there is a NECESSARY universe* in MV, then we have arrived at another problem, which has to do with defining the term necessary. Someone could say, "Sure, maybe there is a necessary universe in MV and so all but (∞ - 1) universes is contingent." Such a necessary universe could not have contingent elements. But if universes are defined as being exhaustively described by the contingent elements they contain, then the Multiverse itself could not contain any necessary elements. What is necessary must exist separately from MV, i.e. the necessary being cannot be an element within MV. This precludes the objection where someone simply redefines what a universe is: "If there is a necessary being that exists, then MV would just come to include that thing." But if inclusion in MV means being a universal set which is itself exhausted by membership of only finite and contingent elements, nothing necessary can be included in the set MV. As a solution to these problems, I ask: **what is not an element in ANY set, but is NECESSARILY a *subset* proper of EVERY set?** It is **∅**. It is just the state of affairs in which the potential to exist *is*. One cannot avoid the intuition that the same equivalence of the foundation of the universe, ∅, has parallels with the human mind. After all, the mind can produce no thought which refers to nothing. ∅ is the most primitive concept we can form. This would make sense if ∅ underlies a universe which is fundamentally self-simulating. Indeed, if we take mental causation seriously, then we must face the fact that we arrive at a problem similar to the Cosmological Problem itself when we consider *some* thoughts. Therefore, the causal explanation for certain thoughts might be thought to refer back to an analogous form of necessary being, namely ∅, or a basic plenum of creative potential, combined with the event of actualization granting Form. We might very well think that mind cannot be explained exhaustively by the brain (though its mental powers can be *locally* explained, i.e. if my brain dies, my mind appears to cease to exist). But if instead we think that brains are not producing, but rather *accessing* the ∅, then we have a concept that looks a great deal like the Logos, an active principle which produces order from chaos. ___

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[–] • 0 pt

I had something typed out, but then I typed something else, and then my computer crashed. It's just as well, since every time I compose a response it's different. So I'll just throw-down these ideas for now.

But what is nothing? ...

In CTMU cosmogony, “nothingness” is informationally defined as zero constraint or pure freedom (unbound telesis or UBT), and the apparent construction of the universe is explained as a self-restriction of this potential. CTMU

Nothing, like anything else, is then a "self-restriction" on the unbound potential afforded by "nothingness". Every last "thing", right down to the last particle or even "nothing" itself, can be understood as a "self simulation" of (n) appearing as (n-*).

The "Holofractal" theory explores the idea that each particle is a simulation of the larger universe, where the "vacuum energy" of each particle is identical to the total energy of the system it inhabits.

Another way to describe "nothingness" is "infinite mass", owing to its boundless form as a mathematical point. Since an Infinite Mass condenses to a point, and since mass = potential, infinite mass = unbound potential. Since it's "infinite", it must have the means of its own realization (sentience), and the means of it's own "non realization" (volitional consciousness).

...the mind can produce no thought which refers to nothing.

A bit of "Zen" grammar resolves things here. Suppose instead we say ...

The mind can produce "no-thought" which refers (directly and without subject/object division) to "nothing(ness)".

I realize this is a cheap Zen "parlor trick", to use terms like "no mind" or "no thought", yet that's literally the issue at hand if we want to understand reality as like an "empty set".

...scientific reduction itself is the description of relations applying over subsets of U, to reduce them to smaller and smaller subsets, where (n-1) is thought to have greater explanatory power than (n).

I've never see a single shred of evidence which supports the notion that (n-1) has greater explanatory power than (n). That's like saying 10-1 is a better description of 10 than 10. It's one thing to talk about science, but there's a thin line between scientific inquiry and the outright dictatorship of terms, and the "Universities" and their Academic mouthpieces are clearly on the dictatorship side of the isle.

If we presume objects in the (n-1) domain are subject to scientific observations, it makes little to zero sense to presume (n) isn't also subject to observations as well, even if we can't properly call them "scientific" observations in the traditional sense of "technologization".

The moment we realize (n) is indeed subject to "realization", the "meaning of life" becomes quite clear, as there's no other apparent means for (n) to be realized other than in the suspension of the very thing which makes (n) unrecognizable: namely the suspension of volition (karma). Volition serves the dual role of concealing the Absolute (n) for the sake of the very possibility of realizing the Absolute upon the suspension of volition (karma). Without concealment, revelation isn't possible.

Upon its realization, the (n) is understood as "empty", like an empty set, in the sense that it has boundary conditions, yet it generates precisely zero information about itself. The realization of "eternally pure stasis" (ie, God's grace, Nirvana) is the identification of (n).

Operations on (n-1...) offer technological advancement, but nothing in the way of "explaining" the (n) ultimately being operated on. This is why we can have satellites and supercomputers, but nothing approaching an explanation of reality. The conflation of science for technology amounts to the supposition and virtual insistence that (n) can't be subject to inquiry. Academic theories of science refer back only to the "university" from whence they originate, not to the "universe" (n) which they're falsely purported to be investigating. Claims of a forthcoming "theory of everything" are pure politics, and made for the same reasons politicians tell lies. The (-1) part of the (n-1) equation is ultimately the "self", which apparently seeks preservation at any cost. The alienation, from virtually everyone, associated with realizing the Absolute (n) is reason enough to avoid going down that rabbit hole; but for some of us, it was our fate.

...

I had another sort of realization just recently, that "relative motion" can't be motion whatsoever. Einstein understood this: whatever appears as "matter" or "motion" is just space-time curled-up and curved in higher dimensions. In fact, "motion" and "relative" can't exist whatsoever without a backdrop of eternal stasis, as there would otherwise be no grounds for "relativity" to proceed.

The appearance of motion is really just the "empty set" (n) operating on itself. Self realization is the realization that "self" is just another way of (n) operating on itself. Ultimately, the "self" which comes to "self realization" is the very (n) we're not supposed to have explanations for.

It's clear from a "dictatorship" POV that individual human subjects realizing (n) is the very antithesis of wielding power over them, and hence we can regard society as one big distraction away form the very sort of self-awareness which would render dictators absolutely powerless.

[–] • 0 pt (edited )

I've never see a single shred of evidence which supports the notion that (n-1) has greater explanatory power than (n).

Yes, I made an error here. I am playing with the cardinality of a set, but I'm doing the impossible of jumping back and forth between numbers and concepts. For example, (n) would be the universe as a set of concept (things with essences), where science reduces that model to (n-x) and reduces the number of concepts within the universe, while actually increasing the number of elements in the set. Concepts decrease, but number of 'particles' increase. I can't do this in one variable, and I'd have to start modeling it as a relation between the particle view of the universe and essentialism, and then I'd have to get into mapping it as ordered pairs onto some space that I don't understand. It would be way beyond me.

It's clear from a "dictatorship" POV that individual human subjects realizing (n) is the very antithesis of wielding power over them, and hence we can regard society as one big distraction away form the very sort of self-awareness which would render dictators absolutely powerless.

I agree completely.

I had another sort of realization just recently, that "relative motion" can't be motion whatsoever. Einstein understood this: whatever appears as "matter" or "motion" is just space-time curled-up and curved in higher dimensions. In fact, "motion" and "relative" can't exist whatsoever without a backdrop of eternal stasis, as there would otherwise be no grounds for "relativity" to proceed.

This is fascinating to me, but I am not sure I completely understand it. If you ever get free time and want to expand on it, make a post or something. I know PS and myself would be interested in this sort of thing. You could even throw it in /LogosRising or the /Philosophy sub.

[–] • 0 pt (edited )

I've had other thoughts about the "empty set" idea, based on CTMU principles.

*The Principle of Attributive (Topological-Descriptive, State-Syntax) Duality Where points belong to sets and lines are relations between points, a form of duality also holds between sets and relations or attributes, and thus between set theory and logic*. ...

Essentially, any containment relationship can be interpreted in two ways: in terms of position with respect to bounding lines or surfaces or hypersurfaces, as in point set topology and its geometric refinements (⊃T), or in terms of descriptive distribution relationships, as in the Venn-diagrammatic grammar of logical substitution (⊃D). ...

Because states express topologically while the syntactic structures of their underlying operators express descriptively, attributive duality is sometimes called state-syntax duality. CTMU

My idea is that "set theory" is related to "point set topology and its geometric refinements", namely as a means of describing "motion (or duration) in space". This gives us a model conforming to our senses, yet senses have been wrong before (eg. Geocentrism). Our senses never changed once the truth (eg. Heliocentrism) became widely realized, just the model we used to interpret them.

Via 21st century standards, the model we need is one describing how and why computation (cognition) is universal, the question begged by today's common interpretations of "objects in space" or a "field". If these objects can be calculated (particularly objects many millions or billions of light years away), then why should calculations about them be a universal process, in advance of any actual calculations performed by scientists, unless the universe is fundamentally cognitive?

Because set theory is directly analogous to "descriptive distribution relationships" aka Venn diagrams, the key to describing reality as an "empty set" is akin to describing it as a Venn diagram containing everything. Since the diagram "contains everything", it appears to contain absolutely nothing, in the sense there are no "descriptive distribution relationships" depicted by it.

*Constructive-Filtrative Duality Any set that can be constructed by adding elements to the space between two brackets can be defined by restriction on the set of all possible sets. Restriction involves the Venn-like superposition of constraints that are subtractive in nature; thus, it is like a subtractive color process involving the stacking of filters. Elements, on the other hand, are additive, and the process of constructing sets is thus additive...* CTMU

So if "Any set that can be constructed by adding elements to the space between two brackets can be defined by restriction on the set of all possible sets.", then can an empty set be defined by zero-restriction on the set of all possible sets?

I think if we define the empty set as "zero-restriction on the set of all possible sets", then we can define the universe as an empty set. Because the empty set can't have any "descriptive distribution relationships", it's empty, yet also because it can't have any such relationships with or among other things, the empty set must have "descriptive distribution relationships" (and point-set topology) as intrinsic compliment to itself, for the "realization" of itself.

You stated...

Such a necessary universe could not have contingent elements. But if universes are defined as being exhaustively described by the contingent elements they contain, then the Multiverse itself could not contain any necessary elements.

I think this is directly related to the "realization" I had about "motion = illusion". Motion appears from a local perspective, but from the global perspective it can't exist. There's nothing for the global-reality to move with respect to. Any apparent motion or duration is always and only with respect to other objects, and no favored perspective of motion exists in "Relativity".

The only "absolute necessity" I can muster is "self containment" which entails "self description" and "self realization", all of which happens regardless if the universe is actually "nothing", or if it's "something" which appears as "nothing" at the global scale.

The universe could just as well exist as "nothing", entirely without "elements" for a virtual eternity and also have elements, but not the other way around. In other words, the universe can't just be "nothing", it must appear as "some things" for the very sake of "self containment", which ultimately entails life and consciousness in the bargain.

The thing about the "empty set" is that its still a set, which mean it can be realized in the very sense requiring conscious, particularly questioning beings like us to ultimately realize what "exist forever" means.

Religions talk about "life after death", but seemingly take this too literally. The religious "death" is allegory for stillness of mind. When stilled, the mind defaults to its eternal state (infinite mass/potential, pure freedom), and only then is the "meaning of life" fully revealed, since "pure freedom" must somehow include its own realization or else it's not eternal nor "reality".

The CTMU addresses many of these issues, as dictated by John Wheeler's prior analysis.

No continuum: The venerable continuum of analysis and mechanics is a mathematical and physical chimera....As Wheeler puts it: “A half-century of development in the sphere of mathematical logic has made it clear that there is no evidence supporting the belief in the existential character of the number continuum.”

No space or time: Again, there is “no such thing at the microscopic level as space or time or spacetime continuum.” ... Wheeler quotes Einstein in a Kantian vein: “Time and space are modes by which we think, and not conditions in which we live”, regarding these modes as derivable from a proper theory of reality as idealized functions of an idealized continuum: “We will not feed time into any deep-reaching account of existence. We must derive time—and time only in the continuum idealization—out of it. Likewise with space.” CTMU (quoting Wheeler quoting Einstein & Kant)

An "infinite mass" suffices for the physical description of the universe, as such a mass has the required potential to "self replicate" in the form of "realizing itself" via cognitive agents (us humans for example).

The "necessary" elements in reality aren't the physical elements per-se, but the "non distinction" holding between objects and observer, namely between reality as the "empty set" and the very observation of that empty set (Nirvana). The contingent elements exist for the sake of the realization of the non-contingent. The non-contingent neither exists nor "not exists".