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1. Easy. Program a computer with the axioms of ZFC. Not powerful enough? Program it with ZFC+Con(ZFC). Repeat as necessary. 2. The Turing machine itself is

1. Easy. Program a computer with the axioms of ZFC. Not powerful enough? Program it with ZFC+Con(ZFC). Repeat as necessary. 2. The Turing machine itself is logically consistent. The semantic interpretation of its data need not be logically consistent: it's possible to write a computer program that prints "2+2=5" to the screen. Likewise, a human brain may be made of physics and physics may be logically consistent, but the semantics of the data in the brain (the "beliefs") need not be consistent. 3. No, no need to reject collapse itself (though I'm more of an Everett guy). Penrose postulates the existence of a new level of physics that is not Quantum Mechanics. People who aren't careful call it an extension of QM, but QM's computing model is limited to solving a small subset of NP problems (BQP) in polynomial time, and BQP is theorized to be a proper superset of P. However, Penrose's proposed physics would allow solving problems outside of NP in polynomial time. Hence, Penrose's physics IS NOT quantum physics. 4. It is physically impossible for computers (as we understand them today) to solve problems outside of the set we call "Turing complete". Penrose claims that there's at least one problem outside that set that the human brain can nonetheless solve: checking the consistency of formal systems. If Penrose's assumption were true, computers (as we understand them today) would consequently be strictly less powerful than the human brain. The crux of Penrose's position is accepting this consequence as true, then asking how that's possible. I reject that hypercomputation of problems beyond Turing-completeness is what's happening, i.e. that humans simply hold unjustified beliefs about the answers to such problems. I do not doubt that Penrose is an intelligent and educated person. However, his expertise is General Relativity. It's bad enough when he tries dipping his toes into Quantum Physics, but when he wades into Physics of Computation, he is no longer acting as an expert, but as an interested layman. It's only because of the authority he holds as a GR expert that his argument is treated seriously, but that authority is not actually relevant to his argument.

manyoso 9y ago
#1 Not easy. Prove it. Show a Turing machine programmed with ZFC that can not be modeled by a Turing machine programmed with PE. You can not do so because Turing machines are universal. The fact that you claim this is easy tells me you do not understand what a Turing machine is. #2 See fundamental level. Penrose claims that what sets us apart from Turing machines is fundamental. But whatever, you've refuted whatever point you wanted to make by saying human's are not logically consistent. Your snark about Penrose not understanding is as empty as whatever point you were trying to make. #3 Cite a paper showing that objective collapse gives these supposed algorithmic speedups or go home. NOTE: objective collapse theories are an active area of research and are not limited to Penrose by any means. Your claim that they are inconsistent with QM begs for evidence. Cite some or stop spreading nonsense. #4 Your answer here is completely void of any context to the question: where Penrose says that microtubules are limited to humans. I take it you concede that he did not say any such thing? You treat Penrose as an idiot missing obvious problems. I think it far more likely that you've misunderstood. Pity your lack of humility might make it impossible for you to understand what he actually argues rather than your strawmen.
- pdkl95 9y ago
  > You can not do so because Turing machines are universal. Just because Turing machines are universal making it possible to write a program that asks a question about another type of Turning machine doesn't mean that question is decidable.
  - manyoso 9y ago
    Show a Turing machine programmed with ZFC axioms that can do something that a Turing machine programmed with PE can not do. If a Turing machine programmed with PE can simulate the Turing machine with ZFC and thus give the same answer, then I state that the program with ZFC is nothing of the sort.
    - pdkl95 9y ago
      The paper[1] I linked to in my other post shows that you cannot perform such a simulation because it is undecidable. (Specifically, BB(N >= 1919) is provably undecidable in ZFC). [1] http://www.scottaaronson.com/blog/?p=2725 http://www.scottaaronson.com/blog/?p=2725
      - manyoso 9y ago
        I read that when it came out, but I don't see how that is relevant to this question. The link provides a program for a Turing machine that probes ZFC. So can you create a Turing machine that probes ZFC, but can't be proven not to halt? The link emphatically says yes! And it also puts some bounds on the number of states necessary to program such a machine. Please show how that paper describes a Turing machine 'programmed with PE' can't simulate a Turing machine 'programmed with ZFC'? Perhaps we are arguing over what 'programmed with PE' or 'programmed with ZFC' means? The parent post seemed to claim that it was possible to construct a physical computer with the axioms of ZFC built-in so to speak. As opposed to one with PE built-in. Which obviously calls into question the parent post understanding of what exactly idealized Turing machines are.
- Chronos 9y ago

manyoso 9y ago

#1 Not easy. Prove it. Show a Turing machine programmed with ZFC that can not be modeled by a Turing machine programmed with PE. You can not do so because Turing machines are universal. The fact that you claim this is easy tells me you do not understand what a Turing machine is. #2 See fundamental level. Penrose claims that what sets us apart from Turing machines is fundamental. But whatever, you've refuted whatever point you wanted to make by saying human's are not logically consistent. Your snark about Penrose not understanding is as empty as whatever point you were trying to make. #3 Cite a paper showing that objective collapse gives these supposed algorithmic speedups or go home. NOTE: objective collapse theories are an active area of research and are not limited to Penrose by any means. Your claim that they are inconsistent with QM begs for evidence. Cite some or stop spreading nonsense. #4 Your answer here is completely void of any context to the question: where Penrose says that microtubules are limited to humans. I take it you concede that he did not say any such thing? You treat Penrose as an idiot missing obvious problems. I think it far more likely that you've misunderstood. Pity your lack of humility might make it impossible for you to understand what he actually argues rather than your strawmen.

pdkl95 9y ago
> You can not do so because Turing machines are universal. Just because Turing machines are universal making it possible to write a program that asks a question about another type of Turning machine doesn't mean that question is decidable.
- manyoso 9y ago
  Show a Turing machine programmed with ZFC axioms that can do something that a Turing machine programmed with PE can not do. If a Turing machine programmed with PE can simulate the Turing machine with ZFC and thus give the same answer, then I state that the program with ZFC is nothing of the sort.
  - pdkl95 9y ago
    The paper[1] I linked to in my other post shows that you cannot perform such a simulation because it is undecidable. (Specifically, BB(N >= 1919) is provably undecidable in ZFC). [1] http://www.scottaaronson.com/blog/?p=2725 http://www.scottaaronson.com/blog/?p=2725
    - manyoso 9y ago
      I read that when it came out, but I don't see how that is relevant to this question. The link provides a program for a Turing machine that probes ZFC. So can you create a Turing machine that probes ZFC, but can't be proven not to halt? The link emphatically says yes! And it also puts some bounds on the number of states necessary to program such a machine. Please show how that paper describes a Turing machine 'programmed with PE' can't simulate a Turing machine 'programmed with ZFC'? Perhaps we are arguing over what 'programmed with PE' or 'programmed with ZFC' means? The parent post seemed to claim that it was possible to construct a physical computer with the axioms of ZFC built-in so to speak. As opposed to one with PE built-in. Which obviously calls into question the parent post understanding of what exactly idealized Turing machines are.
Chronos 9y ago