# Can a perfectly mathematically describable universe exist in a multiverse?

Assuming inter-universal (see here) travel is possible, can a perfectly mathematically describable universe exist inside a multiverse?

If it could, would that mean that the multiverse is mathematically describable as well?

(A perfectly mathematically describable universe would be perfectly predictable as well. Can one predict the future of such a universe when inter-universal travel is possible and the multiverse is not perfectly mathematically describable?)

• This definitely sounds like philosophy to me, not physics. Physics is the study of the behavior of this universe. – Tanner Swett Apr 29 '15 at 16:38
• @TannerSwett: I was not aware that the study of multiverse theory existed beyond Physics. – Vatsal Manot Apr 29 '15 at 16:40
• I wasn't aware that the study of multiverse theory existed within physics. But I'm neither a philosopher nor a physicist. – Tanner Swett Apr 29 '15 at 16:47
• @TannerSwett: Then why post your opinions at all? – Vatsal Manot Apr 29 '15 at 16:49
• Whether a universe is perfectly mathematically describable can not be known using the scientific method. It can, at best, be a philosophical (i.e. religious) assumption. – CuriousOne Apr 29 '15 at 19:36

Disclaimer: Here is my metamathematical take (but read that as a very very very recreational bunch of statements).

Gödel second incompleteness theorem says that the completeness of a logical theory (i.e. that every statement is true, false or undecidable) is not provable within the theory.

However, if you are willing to assume proofs that "are more complex than the theory" (i.e. if you can have a truth complexity of an higher ordinal than the "ordinal complexity" of your system) then consistency can be proved. This is what, roughly speaking, Gentzen has done proving consistency of arithmetics with truth complexity $\varepsilon_0$ (the countable ordinal that is limit of sequences $\{\omega, \omega^\omega, \dotsc,\omega^{\omega^{\dotsc^\omega}},\dotsc\}$, $\omega$ the first infinite ordinal and "complexity" of arithmetics).

So I would say that if an universe is a logical theory, and the multiverse is a more complex logical theory, we may formalize a complete description of the universe by means of "universe statements, with multiverse truth complexity". Nevertheless, we could not show the completeness of the multiverse within itself (Gödel incompleteness).

• Goedel's theorem only applies to logic systems which have infinite sets of propositions and which use quantors that apply to an infinite number of those propositions at once. It would not apply to a finite universe with a large, but finite number of distinguishable physical states. – CuriousOne Apr 29 '15 at 19:41
• @CuriousOne Gödel's theorems apply to any logical system that contains arithmetic (i.e. where you can at least define natural numbers and primitive recursive operations/functions plus you have the quantifiers). And it would be applicable to every logical theory that contains the real numbers from $0$ to $1$ (and it is quite natural that even a finite universe should contain the continuous segment $[0,1]$, for example a continuous interval of time). – yuggib Apr 29 '15 at 20:06
• Any set of propositions over a finite set is a trivial counterexample to Goedel (and it's not covered by Goedel, either, otherwise Goedel would be false). A proposition cal always be proven to be either true of false by simply calculating it for all possible elements of the set. I have yet to see a universe in which an infinite number of physically measurable states exists. Generally speaking, I regard any mention of Goedel and physics in the same book as a clear sign of a pseudoscience approach to reality. That includes otherwise actually intelligent authors. – CuriousOne Apr 29 '15 at 20:12
• Obviously if you theory contains only a finite number of elements you cannot use Gödel theorems. But I have yet to see a physical model that does say that the length of circumference of a circle of radius one (in any unit of measure) is not an irrational number. To define any irrational number, you need all the natural numbers, i.e. a theory that contains arithmetics. In the finite universe that you describe, the real numbers do not exist, so you cannot have the more trivial things, for example, as I already said, that there exist the continuous interval of time $[0,1]$. – yuggib Apr 29 '15 at 20:22
• @CuriousOne And it seems much much much more pseudo-scientific to me such an universe where the length of circumference is a rational number than one where one can imagine the (hypothetical) use Gödel's theorems. – yuggib Apr 29 '15 at 20:23

How can one predict the future of such a universe when inter-universal travel is possible and the multiverse is not perfectly mathematically describable?

Inter-universal travel should NOT be possible if you are speaking of multiverses in the usual sense of inflationary cosmology. The idea of one universe being separate from another hinges on the idea that they are not causally connected. In other words, no event in one universe can ever affect anything in another universe due to the speed of inflation vs the speed of light.

As for this bit:

I must point out, that a perfectly mathematically describable universe should be perfectly predictable as well.

You are going to have to tell us exactly what you mean by "describable" and "predictable" so we can relate them to conventional metamathematical concepts. In mathematics there are plenty of problems that are "undecidable" but I'm not sure if that corresponds more to your definition of "un-describable" or "un-predictable". Further, it's not even known if we (modern human beings) are even working with the most complete/correct form of mathematics. Most of our math (rigorous formulations of calculus, etc..) is derived from the axioms of Zermelo-Fraenkel set theory with the axiom of choice, or ZFC for short. This formulation of set theory (while brilliant) is merely humanity's best attempt so far to formulate and formalize a theory of logic of sets. This then can house things such as the Peano Axioms which give rise to a theory of numbers and so on, but still the truth of the Axioms always remains in question. The fact that we HAVE undecidable statements in our formal systems is reason (I'll avoid the word "evidence" here) enough to suspect our formulation of mathematics may still need refining by future generations. There's always the (dreadful) possibility that homo-sapiens, as sharp as we are, may simply not have the reasoning power to accomplish the sort of "mathematical describability" needed to "describe" a universe in a multiverse.

Sorry for being so meta-mathematical. But it seems important here to bring these things up.

I interpret "perfectly mathematically describable" as a universe which obeys the same physical laws as our own.

The answer is yes. The standard inflationary model allows for multiple, causally disconnected regions to be inflating simultaneously, creating separate universes. There may be some differences in the physical constants of those universes, but they would obey the same physical laws as our own.

• I've updated my question. – Vatsal Manot Apr 29 '15 at 12:00
• I'll leave the answer, but yeah it's no longer very applicable. – levitopher Apr 29 '15 at 12:54

Assuming inter-universal travel is possible, can a perfectly mathematically describable universe exist inside a multiverse?

I would actually say no. Assuming inter-universal travel is possible, there may be sources of randomness coming in from other universes. Namely, the multiverse could influence the universe, making it unpredictable.

For example, you're observing one universe and have built a satisfactory model to predict its behavior. Then some aliens unexpectedly visit that universe from a different one that you weren't observing. Bam, your model is broken.

If it could, would that mean that the multiverse is mathematically describable as well?

Okay, this is where things start to get a little wonky. Per my response to your first question, I don't think you can create a "perfect" mathematical model of anything within an unpredictable system. There's always the chance that the system will do something you don't expect to the thing you're observing.

However, if you have a perfect mathematical model of the system, you intrinsically have a perfect model of anything and everything within the system. Putting this another way, you can have a perfect model of a universe if and only if you have a perfect model of the multiverse it resides in.

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