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No problem here is you proof - although a bit long: 1. THEOREM: Let a semantic frame be defined as Ω = (Σ, R), where Σ is a finite symbol set and R is a fini

No problem here is you proof - although a bit long: 1. THEOREM: Let a semantic frame be defined as Ω = (Σ, R), where Σ is a finite symbol set and R is a finite set of inference rules. Let Ω′ = (Σ′, R′) be a candidate successor frame. Define a frame jump as: Frame Jump Condition: Ω′ extends Ω if Σ′\Σ ≠ ∅ or R′\R ≠ ∅ Let P be a deterministic Turing machine (TM) operating entirely within Ω. Then: Lemma 1 (Symbol Containment): For any output L(P) ⊆ Σ, P cannot emit any σ ∉ Σ. (Whereas Σ = the set of all finite symbol strings in the frame; derivable outputs are formed from Σ under the inference rules R.) Proof Sketch: P’s tape alphabet is fixed to Σ and symbols derived from Σ. By induction, no computation step can introduce a symbol not already in Σ. ∎ 2. APPLICATION: Newton → Special Relativity Let Σᴺ = { t, x, y, z, v, F, m, +, · } (Newtonian Frame) Let Σᴿ = Σᴺ ∪ { c, γ, η(·,·) } (SR Frame) Let φ = “The speed of light is invariant in all inertial frames.” Let Tᴿ be the theory of special relativity. Let Pᴺ be a TM constrained to Σᴺ. By Lemma 1, Pᴺ cannot emit any σ ∉ Σᴺ. But φ ∈ Tᴿ requires σ ∈ Σᴿ \ Σᴺ → Therefore Pᴺ ⊬ φ → Tᴿ ⊈ L(Pᴺ) Thus: Special Relativity cannot be derived from Newtonian physics within its original formal frame. 3. EMPIRICAL CONFLICT Let: Axiom N₁: Galilean transformation (x′ = x − vt, t′ = t) Axiom N₂: Ether model for light speed Data D: Michelson–Morley ⇒ c = const In Ωᴺ, combining N₁ and N₂ with D leads to contradiction. Resolving D requires introducing {c, γ, η(·,·)}, i.e., Σᴿ \ Σᴺ But by Lemma 1: impossible within Pᴺ. -> Frame must be exited to resolve data. 4. FRAME JUMP OBSERVATION Einstein introduced Σᴿ — a new frame with new symbols and transformation rules. He did so without derivation from within Ωᴺ. That constitutes a frame jump. 5. FINALLY A: Einstein created Tᴿ with Σᴿ, where Σᴿ \ Σᴺ ≠ ∅ B: Einstein was human C: Therefore, humans can initiate frame jumps (i.e., generate formal systems containing symbols/rules not computable within the original system). Algorithmic systems (defined by fixed Σ and R) cannot perform frame jumps. But human cognition demonstrably can. QED. BUT: Can Humans COMPUTE those functions? (As you asked) -> Answer: a) No - because frame-jumping is not a computation. It’s a generative act that lies outside the scope of computational derivation. Any attempt to perform frame-jumping by computation would either a) enter a Goedelian paradox (truth unprovable in frame),b) trigger the halting problem , or c) collapse into semantic overload , where symbols become unstable, and inference breaks down. In each case, the cognitive system fails not from error, but from structural constraint. AND: The same constraint exists for human rationality.

vidarh 1y ago
None of this is relevant to what I wrote. If anything, they sugget that you don't understand the argument. If anything, your argument is begging the question - it's a logical fallacy - because your argument rests on humans exceeding the Turing computable, to use human abilities as evidence. But if humans do not exceed the Turing computable, then everything humans can do is evidence that something is Turing computable, and so you can not use human abilities as evidence something isn't Turing computable. And so your reasoning is trivially circular. EDIT: To go into more specific errors, this is fasle: > Let P be a deterministic Turing machine (TM) operating entirely within Ω. > > Then: Lemma 1 (Symbol Containment): For any output L(P) ⊆ Σ, P cannot emit any σ ∉ Σ. P can do so by simulating a TM P' whose alphabet includes σ. This is fundamental to the theory of computability, and holds for any two sets of symbols: You can always handle the larger alphabet by simulating one machine on the other. When your "proof" contains elementary errors like this, it's impossible to take this seriously.
- ICBTheory 1y ago
  You’re flipping the logic. I’m not assuming humans are beyond Turing-computable and then using that to prove that AGI can’t be. I’m saying: here is a provable formal limit for algorithmic systems ->symbolic containment. That’s theorem-level logic. Then I look at real-world examples (Einstein is just one) where new symbols, concepts, and transformation rules appear that were not derivable within the predecessor frame. You can claim, philosophically (!), that “well, humans must be computable, so Einstein’s leap must be too.” Fine. But now you’re asserting that the uncomputable must be computable because humans did it. That’s your circularity, not mine. I don’t claim humans are “super-Turing.” I claim that frame-jumping is not computation. You can still be physical, messy, and bounded .. and generate outside your rational model. That’s all the proof needs.
  - vidarh 1y ago
    No, I'm not flipping the logic. > I’m not assuming humans are beyond Turing-computable and then using that to prove that AGI can’t be. I’m saying: here is a provable formal limit for algorithmic systems ->symbolic containment. That’s theorem-level logic. Any such "proof" is irrelevant unless you can prove that humans can exceed the Turing computable. If humans can't exceed the Turing computable, then any "proof" that shows limits for algoritmic systems that somehow don't apply to humans must inherently be incorrect. And so you're sidestepping the issue. > But now you’re asserting that the uncomputable must be computable because humans did it. No, you're here demonstrating you failed to understand the argument. I'm asserting that you cannot use the fact that humans can do something as proof that humans exceed the Turing computable, because if humans do not exceed the Turing computable said "proof" would still give the same result. As such it does not prove anything. And proving that humans exceed the Turing computable is a necessary precondition for proving AGI impossible. > I don’t claim humans are “super-Turing.” Then your claim to prove AGI can't exist is trivially false. For it to be true, you would need to make that claim, and prove it. That you don't seem to understand this tells me you don't understand the subject. (See also my edit above; your proof also contains elmentary failures to understand Turing machines)

vidarh 1y ago

None of this is relevant to what I wrote. If anything, they sugget that you don't understand the argument. If anything, your argument is begging the question - it's a logical fallacy - because your argument rests on humans exceeding the Turing computable, to use human abilities as evidence. But if humans do not exceed the Turing computable, then everything humans can do is evidence that something is Turing computable, and so you can not use human abilities as evidence something isn't Turing computable. And so your reasoning is trivially circular. EDIT: To go into more specific errors, this is fasle: > Let P be a deterministic Turing machine (TM) operating entirely within Ω. > > Then: Lemma 1 (Symbol Containment): For any output L(P) ⊆ Σ, P cannot emit any σ ∉ Σ. P can do so by simulating a TM P' whose alphabet includes σ. This is fundamental to the theory of computability, and holds for any two sets of symbols: You can always handle the larger alphabet by simulating one machine on the other. When your "proof" contains elementary errors like this, it's impossible to take this seriously.

ICBTheory 1y ago
You’re flipping the logic. I’m not assuming humans are beyond Turing-computable and then using that to prove that AGI can’t be. I’m saying: here is a provable formal limit for algorithmic systems ->symbolic containment. That’s theorem-level logic. Then I look at real-world examples (Einstein is just one) where new symbols, concepts, and transformation rules appear that were not derivable within the predecessor frame. You can claim, philosophically (!), that “well, humans must be computable, so Einstein’s leap must be too.” Fine. But now you’re asserting that the uncomputable must be computable because humans did it. That’s your circularity, not mine. I don’t claim humans are “super-Turing.” I claim that frame-jumping is not computation. You can still be physical, messy, and bounded .. and generate outside your rational model. That’s all the proof needs.
- vidarh 1y ago
  No, I'm not flipping the logic. > I’m not assuming humans are beyond Turing-computable and then using that to prove that AGI can’t be. I’m saying: here is a provable formal limit for algorithmic systems ->symbolic containment. That’s theorem-level logic. Any such "proof" is irrelevant unless you can prove that humans can exceed the Turing computable. If humans can't exceed the Turing computable, then any "proof" that shows limits for algoritmic systems that somehow don't apply to humans must inherently be incorrect. And so you're sidestepping the issue. > But now you’re asserting that the uncomputable must be computable because humans did it. No, you're here demonstrating you failed to understand the argument. I'm asserting that you cannot use the fact that humans can do something as proof that humans exceed the Turing computable, because if humans do not exceed the Turing computable said "proof" would still give the same result. As such it does not prove anything. And proving that humans exceed the Turing computable is a necessary precondition for proving AGI impossible. > I don’t claim humans are “super-Turing.” Then your claim to prove AGI can't exist is trivially false. For it to be true, you would need to make that claim, and prove it. That you don't seem to understand this tells me you don't understand the subject. (See also my edit above; your proof also contains elmentary failures to understand Turing machines)