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I wish to provide some logical relations between the what is physics and what is meta-physics. I don't intend to provide too many examples of either but rather elucidate a general pattern for distinguishing between the two. Consequently, while thinking about this topic I realised that I also have to provide some logical relations between what we believe is the universe and how that relates to the notion of a multiverse.

Let's start with the following basic definition:

(1) The Universe is everything that exists.

But we should also consider:

(2) a universe includes everything that can exist; as well as,

(3) all permutations.

When I use the phrase "that exists" I am specifically talking about all physical processes in our Universe today. It is essentially our reality, but at this point I'm not stating whether our reality is purely describable by universal laws of physics.

The next phrase "everything that can exist" could be interpreted in two ways: (2.1) all physical processes that are consistent with our laws of physics, regardless of whether these processes are observed today or not; or, (2.2) all logically consistent universes with their own set of parameters and physical laws, regardless of whether they exist or not. In both cases there may be the need to add a meta-physical element; however, I will elucidate where such a concern becomes necessary. The addition of meta-physical considerations makes things a little messier. See later.

Finally, (3) is the set of all permutations of universes. This would include universes that are not physically consistent. This last consideration is merely abstract but included for completeness.

Physicality and observability

Definition (1) can be readily related to the idea of an observable universe. Originally I had astronomy in mind, so the observable universe is basically everything that we can see with our telescope, it has some finite spatial extent. This obviously suggests that there is a part of the universe which we can't see but believe it exists due to the consistency of our physical laws. Furthermore, if we were to consider observation in a quantum mechanical sense then there is a limit to what we can 'see' versus what physically exists. In short, the observable universe (U_obs) is a subset of the physical universe (U_phys). As more discoveries are made then the set the information contained in U_obs will increase (as an aside, we could also label U_obs as U_discovered depending on one's preferred usage of the words). The maximum possible size of U_obs is that of U_phys. Consequently, something which is unphysical cannot ever be observed.

To be clearer on this I see the universe as defined in definition (1), which I will denote U_1, has a greater size than U_obs. This universe U_1 is essentially all things that exist regardless of whether we observe them or not. This is the most commonly accepted definition of the Universe and is equivalent to U_physical for most physicists (U_1 = U_phys). This is logical consistent as U_1 is includes everything that exists, which is most likely greater than everything that we have observed. U_1 can certainly not be smaller than U_obs. Also, U_1 can never be smaller than U_phys. It is possible that U_1 could indeed be greater than U_phys if we are to include God, or other metaphysical concepts. In that case, U_1 is a union of U_phys and U_metaphysical. As I said previously, the inclusion of meta-physical elements into the universe makes things messier and, again, I will leave this for later.

Definition (2) addresses some subtleties that are apparent within the notion of a physical universe, and is the basis for the consideration of a multiverse. While definition (1) is restricted to talking about everything that does exist, definition (2) considers the inclusion of processes or objects that currently don't exist but could perhaps exist. An example of (2.1) is the Inflaton field that is suggested to exist during the era of Inflation. In certain models of Inflation this field once existed which means that it should be consistent with our physical laws, but it is no longer present in our current universe. It does not exist now so can't be in U_1, but it did exist, or it might have existed, which places it into U_2.1. I think there is a case for playing with semantics here.

With regards to whether the Inflaton should be included in the definition of U_phys I'd have to admit that I'm hesitant. U_phys can be defined to accept all processes that have provably existed. Once something is proven then it should be wholly contained within U_phys and almost wholly contained in U_obs. Furthermore, we could redefine U_1 to accept all processes that once existed in our universe but are no longer present. These processes are obviously consistent with our physical laws and there is a good case for including them in U_1. That said, there are other possible constructions that we can make. There are other ideas about quantum fields that might have existed, or may exist, which are completely consistent with our physical laws but have never existed and will never be observed to exist. This last sentence is a more general version of (2.1). Whatever these more general constructions are, they cannot be included in our definition U_1. Again we could perhaps suggest that U_phys include these extra constructions since they are consistent with our physical laws regardless of the fact that they don't exist.

Preferentially, I will use the definition that I outlined above: U_1 = U_phys. These are all physical processes and objects that currently exist and provable so. I would also be inclined to include the notion that if the Inflaton is proven to exist, or to have existed once, then it would fit into U_1 and consequently U_phys. Meaning that it would no longer be part of U_2.1.

Following on from this line of thought is definition (2.2) where we consider all possible permutations of physical universes. These are possible universes that need not exist but that can be constructed mathematically or algorithmically. They conform to physical laws which are very similar to our own but the parameters are different. This essentially leads to a notion of a physical universe which is internally consistent but very different from our own. If we can algorithmically generate a universe that adheres to U_2.1 then we should find many similarities between it and our own universe, that is to say that U_1 is a subset of the all possibilities given from the set U_2.1. On the other hand, if we greater change the parameters of (2.1) then we may generate a universe which does not resemble our own universe at all. That is to say that U_2.1 is a subset of U_2.2. Within the definition of U_2.2 I would include the possibility of a universe with physical laws quite different from our own but does not include logical inconsistency.

Multiverse

This finally leads me to a definition of a multiverse:

A multiverse is a union of universes that are consistent with a set of rules and parameters given by definition (2.2).

The first thing to point out is that while U_1 is a particular subset of U_2.2, the existence of either does not guarantee the other. U_1 is our particular universe and from an anthropic line of thinking: we know it exists because I am here to talk about it. The existence of U_1 does not guarantee that a very similar universe exists, but one that is logically disjoint. By necessity of the plurality in the definition above a multiverse requires at least two universes. It does not, however, require that these two universes have the same physical laws nor that these two universes can pass information to each other. This general sense of a multiverse is one that I shall label as a type M_2.2 multiverse.

Clearly, if all information about a given universe is unable to travel to another universe then there is no way of knowing whether the other universe exists. In the jargon of mathematics, the intersection of the two universes is strictly zero. Given that we don't presently know if another universe exists then the notion of a multiverse is currently undecidable. If it turns out that we have some evidence for another universe that co-habits our multiverse we are not necessarily guaranteed that we can say much about it. The logical intersection would be non-zero but it could perhaps be infinitesimal and hence almost inconsequential.

An interesting consideration is a multiverse that contains at least two universes that are alike, say our universe and at least one more that has the exact same physical laws and parameters. The universes could be near identical except they are spatially disjoint. This particular type of multiverse, which I will label with as an M_1 type multiverse, is the type seen on the old Sci-Fi TV series Sliders and is the type seen in StarTrek too when they travel into a mirror universe. The logical intersection may be very low but it is "wide" enough to let some exotic process transfer information from one universe into the other.

This latest figure shows a multiverse M of unstated type which contains our universe U_1 plus six other universes enumerated by the letters A through F. I will use this diagram to make clear each of the terms used above. Universe A is disjoint from all other universes, there is zero overlap. Universes C and D have an infinistessimal overlap, or intersection. While universes E and F have a significant overlap and hence should have a considerable exchange of information.

I also believe that a multiverse of multiverses can be sufficiently re-defined such that it is a single multiverse.

Meta-physics and phenomenology

Now I will return to how the ideas of meta-physics and phenomenology fit into these definitions, and why it adds further complication why it is unpleasant for science. First, I will point out two possible types of phenomenology. One which is a physical phenomenology, that is one which is an unknown in a physical process but is expected to have a physically describable origin. An example of this is mass. At the moment we don't know the real origin of mass: why does it exist and what is its particle of propagation if such a thing exists. When physicists talk of phenomenology they often use the word in this context.

The other type of phenomenology is an invocation of something meta-physical which is the origin of something physical. This is the definition which I prefer as I believe it fits the meaning of the word better. This version of the word fits well with the phrase "deus ex machina"; an explanation of the physical process is not known so we will invent and explanation that does not conform to the laws of physics. An example would be one where God creates a lightning bolt that is observed in our universe. While lightning can be understood from a physics point of view, this was not always the case. Lazily, we could simply assert that God created the lightning bolt and therefore its origin comes from outside our universe. For me, this is pure phenomenology.

I will asset that this second type of phenomenology does not have a place in science because it does not conform to our physical laws. This does not mean that God does not exist, nor that phenomenology is not present in our universe but rather it is not part of our physical universe and hence the understanding of phenomenology is outside of the domain of science. There is no reason to force such phenomenological to conform to logical reasoning: it is meta-logical. Atheists will not (or should not) permit the existence of meta-physics and phenomenology. While meta-physics is not the same as saying that God exists, both fall into the category of phenomenology. Either way, it isn't science.

For theists their morality comes from God and hence has a phenomenological origin. Conversely, for atheists their morality comes from within the physical universe and absolutely not from phenomenological source. This is why theists can assert their morality to be absolute and why atheists should absolutely not permit their morality to be construed as an absolute concept. If morality originates within the physical universe then it is relative to everything else that exists within it, there is no anchor point with which to make it absolute. I don't see this as a problem and I know many atheists would agree but it would seem that not all do. If the atheists are correct about the non-existence of God then they should be absolutely clear where they stand logically in all that they believe in (for lack of a better phrase).

This was well known to me before I formulated the argument using the above prescription of defining what a universe is. My stance on this is to remain ambivalent, unbiased and as objective as possible. Although I do favour morality being a human construct, that is to say that it is non-phenomenological, and hence morality cannot be absolute. Again, this does not necessitate that God does not exist but I am not asserting that morality has to come from God; I have merely asserted that for theists their morality comes from God.

None of the above is necessary for describing a multiverse M_2.2 that adheres to strict physical laws. If there is a meta-physical domain then it is a logical extension to each of the definitions above. The existence of U_1 is a given, this is our universe, but as already argued this does not preclude that any other universe exists. To include meta-physics we could re-define U_1 to include both U_physical and U_meta-physical. Likewise, we can do this for U_2.1, U_2.2 and the various types of multiverse.

U-3

The final piece of the puzzle is to consider all possibilities, both consistent and inconsistent. An infinity of permutations. It does not matter if we call this U_3 or M_3. It is the largest set to which it cannot be a subset. It contains within it all universes and multiverses as subsets, as well as all meta-physical possibilities too. It is most likely useless for most debates so will doubtless be quickly forgotten. For those interested in the study of infinity then this is probably an old idea and is along the lines of the universe of sets in mathematics. And at that I think I will stop.