Multiverse Properties Impossible to Prove or Disprove using Physics

Whatever cosmological multiverse model is eventually accepted by the research community, it will likely have conjectured properties that will be difficult to prove using the techniques of theoretical physics. But will it have easily stated properties that are impossible to prove or disprove using mathematical physics? We explore this question by constructing multiverses that have such unprovable properties. Our multiverses, which could be called lattice multiverses, combinatorial multiverses, or model multiverses, are easier to define than the cosmological multiverse and conceptually easier to understand; they are, however, complex enough to exhibit very deep mathematical properties.

In order to construct a specific, highly intuitive, model, we think of a universe U as a collection of rooms and tunnels with precisely defined additional information added.

Shown in Figure PL is a rectangular lattice or universe with rooms (solid circles) and one-way tunnels (solid arrows). We specify each room by giving its coordinates (x,y) with respect to the axes shown. For any room (x,y), define the lattice-exit distance of that room to be the minimum of x and y.  The lattice-exit distance tells you how close you are to at least one of the two coordinate axes. For example, the lattice-exit distance of room (5,9) is 5.  You are distance 5 units from one of the axes (the vertical axis in this case).  Each room in Figure PL has a number or "label" beside it. A label z on a room (x,y) tells you that, by starting at (x,y) and following one-way passages, it is possible to find a room (x',y') with lattice-exit distance z.  Moreover, z is the minimum such lattice-exit distance to be found in this manner.  For example, the room (5,9) has label 2.  By inspecting Figure PL, you can see that 2 is the minimal lattice-exit distance of any room that can be arrived at by starting at (5,9) (there are two such rooms with minimal lattice-exit distance: (2,6) and (4,2)).

Figure PL

DESCRIPTION OF LIFE IN UNIVERSE PL: Imagine you are an inhabitant of the a universe of type PL (Figure PL), and you are in some room (x,y).  Starting at that room, you must explore by traveling from room to room along one-way tunnels.  The coordinates of each room and its label are posted on the wall of that room. No other information is given. The object of your exploration is to locate a room (x',y') of minimal lattice-exit distance z (specfied by the label z on the start room (x,y)).   If, as it may, z = x or z = y, then the exploration is trivial, ending at the start room (x,y). When the minimal-lattice-exit-distance room is located, you can end the exploration, relax and take a brief break from the routine.  After your break, you are placed at random in another room of the universe. If that room has a label that you have already encountered, skip the exploration, relax and take your break. Otherwise, you repeat the exploration for the new minimal-lattice-exit-distance room.

SIGNIFICANT LABELS:  A label z for a vertex (x,y) is called a significant label for (x,y) if z is smaller than minimum(x,y) (such labels may require a significant amount of exploration).  In Figure PL, insignificant labels are in parentheses while significant labels are not.  Given a multiverse M, defined below, we shall ask if that multiverse has arbitarily large universes U with nice structure ("cube - like") in which the number of different significant labels is very small. After a long life, a citizen of such a universe when relocated at random would almost certainly end up at an (x,y) with either an insignificant label or a significant label previously encountered -- lots of enjoyable breaks punctuated by an occasional adventure. Heaven, so to speak!

(1) Some of the inhabitants of a universe such as Figure PL might wonder how the labels (or data structure) got added to the rooms and how much work was involved.  For large complexes of rooms and tunnels the work could be considerable.  On the other hand, since the exploration is going to be repeated over and over again for generations, the work of adding the labels might be worth it. In a certain sense, the nature of the labels allowed at the rooms and, possibly, tunnels is the "physics" of a universe.

(2) Our previous discussion suggests that it would be interesting to know something about the number of different significant labels.  In Figure PL, the set of significant labels is {1, 2, 7}.  There are just three different significant labels.  Another thing that would be useful to know is how many rooms have significant labels (7 rooms in our example). We are going to focus on the former question as the universes U range over all of M, making properties of significant labels a multiverse issue, not just an issue for a particular universe.

EXAMPLES AND SIGNIFICANT LABEL THEOREMS FOR MULTIVERSES:

Start with a directed graph M that has vertex set the Cartesian product N^2 = N x N where N is the set of nonnegative integers. A multiverse M of the graph M consists of two things:

(1) the set of all induced subgraphs U = M[D] where D is a finite subset of N x N. For example, in Figure PL, the dashed edge from (10,6) to (16,0) represents an edge of M not in M(D).

(2) for each such induced subgraph M[D], a rule for labeling its vertices and (perhaps) edges.

The analogous definition of M will be used when M has vertex set N^k (k-ary Cartesian product).

We consider three different types of labeling rules, PL (path-based labels), TL (terminal-path-based labels), and SL (selection-function-based labels), each a bit more complex than the previous. Each labeling rule will gives rise to its own class of multiverses and to interesting questions about the associated "significant labels." In particular, we show that for each labeling rule and multiverse there is always some universe with very nice cubical geometric structure and few significant labels. We call these results "large-cube theorems."

Only our large-cube theorems about multiverses defined from PL are proved using the mathematics of the sort used in physics and standard mathematics. Our proofs about the structure of the multiverses M defined from TL and SL use mathematics beyond the standard ZFC Axioms (Zermelo, Fraenkel, Choice). The type of math used by physicists (all ZFC math) could possibly be used to prove our large-cube theorems about TL, but no such proof is known. Our large-cube theorems about SL, however, provably cannot be proved using just ZFC mathematics. The fact that we find ourselves in such a state for our lattice multiverses is a sign of the ease with which any of the infinite multiverses proposed by physicists might generate interesting, concrete statements that will be out of reach of string theory, quantum gravity, or any other mathematical theory being developed by physicists.

MULTIVERSES TL and SL:  We give a mathematical presentation of the foundationally interesting multiverses, TL and SL here: (Lattice Multiverse Models).

MULTIVERSE PL:  Here is the ZFC-provable version of our large-cube theorems: (Multiverse PL, large-cube theorem). This theorem can be proved in the framework of "normal" math.  The rule for labeling the vertices in Multiverse PL is described in connection with our discussion of Figure PL above. The ZFC theorem used is a version of Ramsey's Theorem.

What situation should we look for in a cosmological multiverse M that would give rise to properties unprovable in physics? Basically, we need to be wary of assertions about the existence of universes (or substructures within universes) of M with exceptional properties with respect to all universes in M.

REFERENCES:

(1) The paper Lattice Multiverse Models , by S. Gill Williamson, proves "large-cube theorems" for lattice multiverse models. For some of these theorems, the only known proofs use (but may not require) theorems independent of ZFC. For others, the proofs require the use of theorems independent of ZFC. The ZFC-independent results in this paper use Harvey Friedman's Jump-free Theorem and depend on his extensive work on the applications of large cardinals to graph theory (see reference (2) below).

(2) The most comprehensive work on combinatorial results whose proofs require the use of large cardinals can be found in the deep and important work of the logician, Harvey Friedman.  In terms of lattice graphs and the material we have been discussing here, his most relevant paper is Applications of Large Cardinals to Graph Theory, particularly Section 4, page 20 on.  This paper contains a statement of Friedman's Jump-Free Theorem which is the key result we need to prove the significant label theorem for lattice multiverses TL and SL.

(3) The paper Large-Scale Regularities of Lattice Embeddings of Posets , by Jeffrey B. Remmel and S. Gill Williamson, explores applications of classical Ramsey Theory and the Jump-Free Theorem to the structure of partially ordered sets (posets) on lattices.

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