On Monday, January 13, 2025 at 11:58:38 PM UTC+1 Jesse Mazer wrote:
Doesn't Godel's theorem only apply to systems whose output can be mapped to judgments about the truth-value of propositions in first-order arithmetic? A cellular automaton would seem to have "evolving quantities and/or qualities through numerical or some other equivalent formalism's means", but Godel's theorem places no limitations on our ability to compute the behavior of the cellular automaton for N time-increments, for any finite value of N, so I would think Godel's theorem would likewise place no limitations on our ability to compute the physical evolution of the universe's state for any finite time interval. For some cellular automata it may be possible to set up the initial state so that the question of whether some theorem is ever proved true or false by the Peano axioms (or other axioms for arithmetic) is equivalent to a question about whether the automaton ever arrives at a certain configuration of cells, so Godel's theorem may imply limits on our ability to answer such questions, but this is a question about whether something happens in an infinite time period. I assume there are similar limitations on our ability to determine whether certain physical states will ever occur in an infinite future (straightforwardly if we build a physical machine that derives theorems from the Peano axioms, or a machine that derives conclusions about whether various Turing programs halt), but most of what physicists do is concerned with predictions over finite time intervals, I don't see how Godel's theorem would pose any fundamental obstacles to doing that. Jesse Gödel’s first incompleteness theorem states that any sufficiently strong formal system (capable of arithmetic) contains statements that are undecidable—neither provable nor disprovable within that system. The second theorem says such a system cannot prove its own consistency from its own axioms. These are statements about provability in formal theories. They are not directly about whether you can compute a finite number of steps in a system like a cellular automaton. You can absolutely compute, step by step, the evolving states of a (finite) cellular automaton for N time steps, and Gödel’s theorems do not say you can’t. They say something deeper: if your formal axioms are strong enough to represent integer arithmetic (like Peano Arithmetic or any Turing-complete formulation), there will be statements expressible within that framework which it cannot resolve. That’s a statement about what can or cannot be proven within the system, not about whether a machine can run a simulation for some finite time. You also assume that Gödel’s incompleteness only restricts what can happen in an “infinite time” scenario. For example: “Well, sure, there might be some question about whether a certain configuration arises eventually, but for finite intervals we have no Gödel-limited obstacles.” This misreads Gödel: Gödel’s first theorem does not hinge on infinite time steps; it is about the intrinsic logical structure of the formal system. Even for trivial seeming statements involving finite objects (e.g., “this specific integer has property P”), the theorem shows there can be statements that the system cannot prove or disprove. It’s not that you can’t “run the simulation long enough,” but rather that the theory itself cannot settle certain propositions at all. Yes, questions about infinite evolution (like “does the automaton ever reach configuration C at some unbounded time?”) can become unanswerable in principle if they encode the halting problem or an arithmetic statement. But Gödel’s point is more general: the existence of some undecidable statements is guaranteed, quite apart from whether they manifest in an infinite-time scenario or not. If a cellular automaton is known to be Turing-complete (i.e., it can replicate the behavior of a universal Turing machine), then in principle it inherits exactly the logical limitations that come from being universal. That means there will be statements about the automaton’s global behavior (for instance, whether it will ever reach a certain state from a certain initial condition) that are undecidable. This “undecidability” is not about lacking the ability to compute it up to step N—rather, it’s about no possible proof or disproof existing within a certain formal system’s axioms regarding some carefully constructed properties of that automaton. Put another way, if the theory describing the automaton and its states is as powerful as something like Peano Arithmetic, then Gödel’s result shows there exist statements about that automaton’s behavior that the theory can’t settle. This doesn’t contradict our practical ability to simulate the automaton for finite steps. It simply highlights that some high-level questions (like a halting-like problem for the automaton) could be undecidable in the strong sense Gödel identified. A final misconception is that the existence of straightforward computable truths (e.g., “2 + 2 = 4” or “state s follows from state s_0 after N steps”) somehow negates Gödel’s theorems. It does not. Gödel never claimed “no statement is decidable,” but rather that “there exist some statements that are undecidable.” The presence of plenty of decidable truths or effectively computable processes is not an escape hatch from incompleteness. A formal system can easily prove 2 + 2 = 4, yet still fail to prove or disprove a Gödel-type sentence. Therefore, you seem to mix up “we can compute local steps easily” with “therefore there’s no deeper limit on provability.” A huge jump. Incompleteness is about the existence of certain statements the theory cannot decide at all—not about whether we can check how a system evolves over N steps. It can help to see Gödel’s message as a reminder that once you allow arithmetic (or an equivalent notion of proof) into your theoretical framework, you automatically inherit logical “gaps” you cannot fill from within that framework. Consequently, every time a new observation or phenomenon appears to contradicts some current theory of the universe in the news feeds—be it quantum mechanical or cosmological—standard practice is to say, “We’ll just extend or refine the system,” hoping that, in principle, you can patch all holes. From a Gödelian perspective, however, such hopes are naive, even if made by top researchers constantly; you cannot achieve a neat, airtight completeness once you’re in the realm of arithmetic-level expressiveness. Each new theory you devise may bear fruit—often very real, practical fruit—and it may rectify certain anomalies, but it cannot be the final word, because no single formal system can be both consistent and complete about everything you can say in it. That’s why it’s no surprise to someone with a minimum of familiarity in Gödel’s work that the mind spawns endless re-interpretations and frameworks—especially in quantum mechanics—that run counter to some “standard” stance. They’re not necessarily wrong or right in a final sense; rather, they’re new attempts at formalizing a larger or more nuanced slice of reality. As Gödel implies, there is an unbounded supply of possible formal systems you can keep proposing, and many do yield fresh insights or engineering progress. But none will eliminate the fundamental incompleteness that arises once you assume enough structure to reason arithmetically. So the cycle of “discover contradiction → revise theory → see new contradictions” is essentially built into the logical fabric of how we formalize our knowledge—Gödel merely gave us a precise way to see why that cycle never fully ends. This is also why a ToE, when reliant on evolving quantities/qualities and including physics, has to be “informal” with Gödel in mind. And that’s what this list got so wrong with so many years of splitting hairs with Bruno: informal reasoning can be rigorous. And regarding ToE post Gödel, it has to be that way! All the attacks on Bruno of the “You can’t be serious!” or “that is not a rigorous argument/proof”, or “how can informal thought experiments be taken seriously” - sort just miss the point/lack familiarity with Gödel. I hope this provides some clarity. I’ll have to ask forgiveness for not being able to participate in discussions and reply as much as I’d like but I have too many time restraints atm and not enough time to even read. Hence, my call for either some moderation, more restraint, or whatever is needed to have a more organized list focused more on the ToE topic, instead of spoon feeding, bad faith arguments, or troll engagement/permissiveness. Yes, of course we can talk Trump politics, but not in some trivial "caveman truth" asserting way. I hope you understand. In closing, I suggest looking at the work (Gödel) and making up your own mind to decide for yourself whether Gödel’s theorems apply or not, so you don’t have to take my word for it, as you shouldn't. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To view this discussion visit https://groups.google.com/d/msgid/everything-list/3e559b07-1b9d-4682-9642-c1eaf1d358f4n%40googlegroups.com.
