Physicist Max Tegmark is widely known for proposing that there is a multiverse where mathematical structures would exist as real and actual universes (https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis)

He has suggested that his mathematical structures would be completely consistent. He uses Hilbert's definition of mathematical existence (Which basically says that as long as a mathematical structure is consistent, it will exist)

But he published a relatively recent paper (https://arxiv.org/pdf/0704.0646.pdf) where he says:

I hypothesize that only computable and decidable (in Gödel’s sense) structures exist

This confused me a lot since Gödel's incompleteness theorems deduce that undecidability implies consistency, and decidability implies inconsistency.

So what is happening here? Is he proposing inconsistent mathematical structures as existent now? Did he change his mind?

This confusion gets worse at the end of the article, where he said:

According to the CUH, the mathematical structure that is our universe is computable and hence well-defined in the strong sense that all its relations can be computed. There are thus no physical aspects of our universe that are uncomputable/undecidable, eliminating the above mentioned concern that Gödel’s work makes it somehow incomplete or inconsistent.

This seems to contradict Gödel's theorems: If a universe would be completely decidable defined and complete, wouldn't that mean that it would be inconsistent?

  • 2
    As I understand it, that means he excludes from consideration any structures that are subject to incompleteness. Recall that ‎Gödel's first incompleteness says that any axiomatic system that can represent the arithmetic of the natural numbers must be incomplete or inconsistent. If you throw out the pesky axiom of induction, you no longer satisfy the hypotheses.
    – user4894
    Jun 19 '19 at 4:02
  • 2
    2) Consider for example all the programs you could possibly run on an actual physical computer like your laptop. It's got bounded capacity, hence can't implement induction. So any mathematical structure that could be computed on your laptop is ok. This is of course a FAR cry from "the universe is a mathematical structure." It represents a substantial retreat IMO. And it's almost certainly false. For one thing it's inconsistent with known physics.
    – user4894
    Jun 19 '19 at 4:02
  • 1
    "It excludes much of the landscape of mathematical structures, not to mention that pretty much every successful physical theory so far would violate CUH". He can be interpreted as allowing only finite (but perhaps very large) structures, and treating the "infinite" ones we use as asymptotic approximations. If the model is finite the theory describing it is both consistent and complete (e.g. Boolean algebra).
    – Conifold
    Jun 19 '19 at 15:37

This CUH is just as incoherent as MUH (i.e. "all mathematical structures exists"), even though not as obviously. For "all and only computably decidable mathematical structures exist" to be a coherent claim, one must assume that the notion of "computable structure" is well-defined and absolute. However, "computable structure" cannot be defined without assuming something more or less equivalent to existence of a model N of TC or PA. But Th(N) is undecidable for any such N, as a trivial consequence of Godel's incompleteness theorem!! Hence the whole idea falls flat on its face.

  • Do you think it would still be incoherent if it were modified so that in place of Turing machine computability, it talked about "computability" by some well-defined oracle machine that was at least powerful enough to determine if any WFF in first-order arithmetic was true or false in true arithmetic?
    – Hypnosifl
    Dec 18 '21 at 17:50
  • @Hypnosifl: Interesting question. Let me think about it before getting back to you. =)
    – user21820
    Dec 18 '21 at 17:51
  • @Hypnosifl: I think your modified NCUH is not obviously incoherent, but it is not absolute. It must assume the existence of a single countable model ℕ of PA− that we cannot pin down, and then assert that all and only structures that are decidable relative to (membership in) T = Th(ℕ), equivalently relative to the ω-th Turing jump, exist. But there is a deeper problem now. Let W be the collection of all T-computable programs P such that P represents some well-ordering f(P) on a subset of ℕ. Then W is countable and every member of W exists by NCUH. [cont]
    – user21820
    Dec 18 '21 at 19:35
  • [cont] It turns out that ATR0 suffices to prove that every pair from W represents compatible well-orderings (i.e. one embeds into the other). Now define ◁ on W as the strict-embedding relation, which respects isomorphism ≅. Let V ⊆ W such that every P∈W satisfies f(P) ≅ f(Q) for some unique Q∈V with Q≤P. (V represents W/≅.) Then ◁ is a well-ordering on V ⊆ ℕ! So no R∈W represents ◁↾V, otherwise ◁↾V ≅ f(R) ≅ ◁↾V[◁R], yielding contradiction. Thus if we assert NCUH, then the structure ⟨V,◁↾V⟩ cannot exist, so we must deny the relatively simple mathematics (a bit beyond ATR0) used to construct V.
    – user21820
    Dec 18 '21 at 19:38

An often neglected, but crucial, hypothesis of Gödel's incompleteness theorem is that the theory in question is sufficiently strong — it suffices for the theory to be able to discuss the theory of integer arithmetic.

A standard example of a decidable theory is (Tarski's axiomatization of) Euclidean geometry. Or basically the same thing, the first-order theory of real number arithmetic (called a "real closed field").

(the assertion that the theory of real number arithmetic cannot discuss the theory of integer arithmetic may be surprising; the point, though, is that we aren't asking how to compute operations, but how to discuss questions like whether certain equations have integer solutions. In the first-order theory of real arithmetic, you can't define what it means to be an integer, and thus while you can ask whether certain equations have a solution, you can't ask whether any of the solutions are integers)

  • What you say is true but CUH is still incoherent for a different reason.
    – user21820
    Dec 18 '21 at 6:01

Hilbert was a formalist. He did not mean that merely because a formal system was consistent then the things that the symbols referred to actually exist. He merely took it to mean that the formal system may be worked with and stillgive useful answers.

The position that you ascribe to Hilbert ia called mathematical realism: that consistent mathematical objects actually exist.

Godels incompleteness theorems did not say that undecidabilty implies consistency. But merely that decidability implies inconsistency. And this in any theory that supports Peanos axioms for arithmetic.

Elementary Euclidean geometry, which is Euclidean geometry married to first-order logic without any set theory can be shown to be complete and consistent. This was done by Tarski in 1951. And we can deduce from this that it is weaker system than arithmetic.

Also, the theory of real-closed fields is also complete. This was also accomplished by Tarski in 1940. It turns out that this is related to the previous example since elementary Euclidean geometry is a model of some real-closed field.

I'm not a big fan of Tegmark's theories - so I'll stop here.

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