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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 (edited )

I did recognize your reference to potential within the original post, but this word nothing was thrown around so much I feel as if that concept was being made synonymous with the empty set. I also thought you were saying zero and nothing were distinct because zero possessed something nothing did not, but then I thought you were saying that it is nothing, not zero, that constitutes the empty set. It may have just been hasty reading on my part; time was short last night.

I do recall you differentiating between nothing and the empty set, since the empty set has at least as much being as to be able to contain other being, but it also seemed somehow you were equating them.

An empty set only has being insofar as it has the capacity to receive other being. This is indeed precisely what a type of being without any act would be like - pure potentiae, prime matter, being awaiting act.

But this is still distinct from necessary being, which is the opposite of prime matter, but rather is pure act.

I guess what disturbed me about your post was your association of the empty set with mind, and your description of Mind as the necessary being. That sounds a lot like God, but nothing could be further from God than prime matter.

Aristotle believed in the eternity both of Pure Act - the Prime Mover he called it - and of the world itself, particularly prime matter. And he probably believed it for similar reasons to those you've presented; the Prime Mover required existing material to move and give form to.

But the Church rejects the eternity of the world because it goes against the creatio ex nihilo dogma. Only God is eternal, everything else is created. But then again, we must remember that the Church exists to guide the flock, and human beings in our fallen nature are so sinful that, if the Church did not condemn the eternity of the world, people would turn to worshipping idols, thinking them "divine".

This is not to say the Church dogmatized a lie; that's impossible. Rather, the Church, in forming her doctrines, must work with the limitations of human language. The average Joe won't recognize the difference between "the world" (which has actual being, as it exists now and as we know it) being eternal, and "prime matter", pure potentiae, being eternal. If we call prime matter just that potential for God to create, the empty set itself, then perhaps we can get away with that without contradicting the Church. But I'm wary of such claims, because even that suggests God created creatures, in a way, from Himsrlf, which the Church condemns. Then again, as much must necessarily be true, at the right level of consideration.

And as Smith notes, even Genesis refers to God "moving over the waters" at the beginning. What waters? And why water? Is it because water is receptive to any form? Is this passage then referring to prime matter, to an empty set?

I always sense a danger more risky than its worth in discussing these issues.

Genesis 1:2 (biblegateway.com) - this is definitely the verse to consider with respect to your post. Second verse in the Bible. Not only does God move over the waters (before creating the oceans?), but the earth (before creating the earth?) is empty and void. Does this not describe an empty set?

@KingOfWhiteAmerica

[–] • 0 pt (edited )

But this is still distinct from necessary being, which is the opposite of prime matter, but rather is pure act.

I think you are correct to criticize me here. I made some confused statements toward the end in this regard.

I guess what disturbed me about your post was your association of the empty set with mind, and your description of Mind as the necessary being. That sounds a lot like God, but nothing could be further from God than prime matter.

Yes, exactly. I should have thought about it more before trying to connect the entire argument to mind. Instead, I'd rather say that the image of trinity is reflected in mind, and what the empty set reflects is the 'bottomness' of mind (the most naked thought we can have), which itself cannot come without at least some act - we bracket nothing {...}. To use a metaphor, the human mind is never flat-footed, it is always standing on 'tip toes', with some rudimentary tension in the muscles.

Somehow, I believe the image of God as triune is reflected in the human mind, and the empty set is just the cognitive reflection of the prime matter.

And as Smith notes, even Genesis refers to God "moving over the waters" at the beginning. What waters? And why water?

Precisely. I was inspired a while ago by this imagery. The ocean is a symbol of deep power and potential.

But the Church rejects the eternity of the world because it goes against the creatio ex nihilo dogma.

Yes, I think again we have arrived at one of these points of intersection where I'm interested in the esoteric understanding of reality, but you are careful to behave like a compass and keep me situated so as to always distinguish between these and the teachings of the Church.

If God is all, there is no logical way to get around the fact that the prime matter must be an aspect of God. I think you are clever to point out that (at least on a first analysis) we can distinguish between the prime matter and what we find in the created world. This is the mystery of the emanation, and where the Christian angelic hierarchy (or the Jewish Tree of Life) enters the picture symbolically. There is always the emphasis on making the distinction between the cruder matter of this world and the spiritual reality of more 'divine stuff'.

Genesis 1:2 - this is definitely the verse to consider with respect to your post. Second verse in the Bible. Not only does God move over the waters (before creating the oceans?), but the earth (before creating the earth?) is empty and void. Does this not describe an empty set?

YES! YES! That's it. Scripture had it. This got me super excited.

@KingOfWhiteAmerica