Multiverse Properties Impossible to Prove or Disprove using Physics

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S. Gill Williamson

Whatever cosmological multiverse (i.e., multiple universe) 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 or combinatorial multiverses, are easier to define than the cosmological multiverse and conceptually easier to understand; they do, however, exhibit very deep mathematical properties. To go straight to the mathematical discussion click here.

An Elementary Discussion (very little mathematics)

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 consisting of rooms (solid circles) and one-way tunnels (solid arrows, direction of arrow indicates direction of tunnel). 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 tunnels, 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). The minimal lattice-exit distance rooms reachable from (5,9) are the rooms (2,6) and (4,2), both with lattice-exit distance 2 (in parentheses for reasons discussed later). We refer to this system of rooms, tunnels, and labels as a universe of type PL (Universe PL).

Figure PL

Let's add some life to 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 your universe 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 enjoy a brief break from life's dreary 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 and the good life: A label z for a vertex (x,y) is called a significant label for (x,y) if z is smaller than the minimum coordinate of (x,y). Such a label might require a "significant" amount of exploration by a life form in Universe PL placed at (x,y). In Figure PL, insignificant labels are in parentheses while significant labels are not. Note that the set of significant labels is {1, 2, 7} in Figure PL. There are just three significant labels. Note also the difference between the set of significant labels and the set of rooms that have a significant label. There are 7 rooms that have a significant label in our example. If we imagine a very large PL-type universe with very few significant labels, then the life forms there would visit lots of rooms, take lots of breaks, and do very little work exploring -- the good life!

What we will show: In terms of our discussion above, our program is to define and analyze multiverses of type PL, where each such multiverse is an infinite collection of PL-type universes. Similarly we are going to define "TL-type" multiverses and "SL-type" multiverses. For each multiverse, we will prove a theorem that states that there are universes with arbitrarily large, nicely shaped ("large cubical") subregions with very few significant labels. If our inhabitants were to live in these large cubical subregions, they would end up leading good lives. Our large-cube theorem for the case PL is proved using standard mathematics (like that used in physics). For TL and SL we use nonstandard math (not used in physics). In addition, for SL we prove we must use nonstandard math (physics can't be used).

Multiverses and Large Cube Theorems (a solid discrete math course needed)

For the multiverses TL and SL with references to the important work of Harvey Friedman, see the article Lattice Multiverse Models.

For the mathematics of the multiverse PL see Proof of Large-cube Theorems for Multiverse PL ( pdf). The label function fL used in this discussion is that described in Figure PL above.

"Physics and Geometry" for Multiverse TL

Jump - Free "Light Cone" for TL and SL

"Physics and Geometry" for Multiverse SL