On Tuesday, January 14, 2025 at 9:06:29 PM UTC+1 Jesse Mazer wrote:
On Tue, Jan 14, 2025 at 4:56 AM PGC <[email protected]> wrote: 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. I think you are misunderstanding my claim, I didn't say that Godel's theorem itself is directly stated in terms of number of time steps of some computation, only that if we look for applications of Godel to computational dynamical systems like cellular automata or computable physics (and Deutsch's result on p. 11-13 at https://www.daviddeutsch.org.uk/wp-content/deutsch85.pdf suggests the evolution of any finite quantum system is computable), the only applications will be to questions about whether the dynamical system ever reaches a certain state in an unlimited time. You said yourself that Godel's theorem places no limits on our ability to compute the behavior of such a system for N time steps given any specific value of N, so I don't think you disagree with this. If you do disagree, i.e. you think there is a way of applying Godel's theorem to a finite computable system that places limits on our ability to deduce something about its behavior over a finite series of time steps, please give an example. First I'll address the rest of your post as there's not really much to talk about: Uncomputable inference rules (like the ω-rule) aren’t used in standard physical theories much, so invoking them misses the core point about Gödel’s incompleteness unless you have some incredible non-standard result to show; in which case, prove/show it. Again, we can simulate a finite system for N time steps, but Gödel’s result is not about whether you can brute-force a finite trajectory—it’s about the existence of statements in a sufficiently strong formal framework (one that encodes arithmetic) which no consistent axiom system can decide. Consequently, if a “theory of everything” in physics is robust enough to interpret integer arithmetic, then Gödel’s incompleteness theorems apply. There's really nothing more I can say regarding all the vagueness in your reply, because you clearly are performing rhetorical moves to limit the generality of Gödel's contributions and separating "modern physics" from it. And doing the same while demanding “an example” for finite steps is meaningless unless we specify exactly which formal system’s provability we’re talking about— with the entire range of standard specifications that the question omits, as if we were adding salt to a dish or something. That omission reveals a misunderstanding of Gödel: he never claimed you can’t compute discrete steps in a small system, but rather that any consistent, arithmetic-level theory remains incomplete about some statements. This conflation of “finite-step computation” with “formal provability” underscores why your rhetorical moves lack any sort of precision and ultimately misrepresent/misunderstand not only Gödel’s theorems but the nuanced notion of provability in basic terms. Because provability is relative and computability is not. Your stock fell for me with this reply. Please don't pretend to spoon feed me. You can play teacher with AG, which is out-of-topic regarding ToE. AG can pay for lessons somewhere and play "not convinced" twirling his moustache and adjusting his monocle elsewhere and the trolling/grandstanding, playing god is so abundant... moderate it guys. Passivity kills freedom and discourse. -- 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 [email protected]. 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