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Dasgupta UMBC, 1000 Hilltop Circle, Baltimore, Maryland 21250 A New Approach Refuting Penrose’s Gödelian Argument Arghya Dasgupta UMBC, 1000 Hilltop Circle, Baltimore, Maryland 21250 Abstract Background Related Work Possible Approaches Sir Roger Penrose, though one of the greatest minds in pure In the case of Peano arithmetic or any familiar explicitly Hilary Putnam, G.Boolos and others have already pointed out to know this, we must also prove (or know) that ∀e ∀x(A(e, x) mathematics and physics, is known for his insistence in using axiomatized theory T, it is possible to define the consistency "Con some of Penrose’s technical errors, and he has conceded several ⇒ Φ(e, x)↑ That is, we have to know that A is a sound method Gödel’s incompleteness theorems to support a non-computationist (T)" of T in terms of the non-existence of a number with a certain of the technical points, but there does not seem to be a complete for deciding that our own computations fail to halt: whenever it and even a non-algorithmic view point in studying consciousness. property, as follows: "there does not exist an integer coding a statement of the fundamental problems with this kind of argument. claims that a computation is non-terminating, then in fact that Repeatedly, in books like “The Emperor’s New Mind” (1989), sequence of sentences, such that each sentence is either one of Benacerraf (1967) made two points against the original Lucas computation does not terminate. An alternative resolution of the “Shadows of the Mind” (1994) and “The Large, the Small and the the (canonical) axioms of T, a logical axiom, or an immediate argument, both of which we will develop more precisely here. The contradiction is therefore that a human thinker like Penrose (or, Human Mind” (1997), Penrose has used newer and more consequence of preceding sentences according to the rules of first focuses on the fact that in order to know the truth of the un- perhaps, the entire human race) is unable to establish the thorough versions of incompleteness theorem to support his inference of first-order logic, and such that the last sentence is a provable “Gödel sentence”, one has to know that the formal soundness of the algorithm A that embodies his mathematical views, which were first advanced by the philosopher John Lucas contradiction". However, for arbitrary T there is no canonical system is consistent. The thesis that we humans, unlike the formal abilities. Penrose apparently considers this ridiculous, but I will try (1961). There have been many criticisms to this approach, notably choice for Con(T). system, can know the truth of the system’s un-provable sentence to show that it is more plausible than it might at first seem. by G.Boolos (1989), H.Putnam (1995). But all of them, in essence, is central to the argument. An alternative conclusion, therefore, is deal with the more technical aspects of Penrose’s proof, to which Penrose’s Adaptation of the Incompleteness Theorems that humans may be unable to know that they are consistent. Naturally, the force of this technical objection relies on the human Penrose reacts by coming up with a new version of the proof Penrose’s key idea is essentially the same as that of the Penrose, like many who think his argument persuasive, finds this notion of soundness being the same sort of thing as that of a which addresses the criticisms which are of a syntactic rather than philosopher J.R. Lucas in, an article which has attracted simply ridiculous; but I will try to show that it is more plausible than machine, namely something describable in a computable way. the semantic kind. But the core of his view never changes. In my responses from several writers over the last thirty-five years. In it may seem at first sight (for example, we can find inconsistencies Penrose, it seems, believes that we have access to an intuitive research, I am trying to find a more general problem with the brief, Lucas argued as follows. Gödel’s incompleteness theorem in Penrose’s own published opinions) and does not have the dire notion of soundness which avoids the limitations of the machine application of Gödel’s theorem with regard to consciousness. In shows that, given any formal system, there is a true sentence intellectual consequences that he fears. notion, yet can still be used freely in reasoning about more fact, I am going to claim that it might not even be justified to apply which the formal system cannot prove to be true. But since the Benacerraf’s second point is that Lucas equivocates on the notion precisely specified notions of soundness. This is the reason we Gödel’s work to consciousness and the mind in the way Penrose truth of this un-provable sentence is proved as part of the of proof. Once “proof for humans” is given a clear definition, the need the second objection to Penrose’s claims. I wish to more and Lucas is used to. incompleteness theorem, humans cue prove the sentence in argument applies to humans as well. This point was repeated fully exhibit the fact that any such argument applies just as well to question. Hence, human abilities cannot be captured by formal against Penrose by both Boolos and Putnam, and it can be made human thinkers as it does to machines. It uses the “formality” of Background systems. again in more detail, to show how its assumptions are very weak the system only to establish that the set of algorithms is and plausible. While Penrose believes himself to have overcome enumerable; but the set of possible mathematical sentences, and Gödel’s Incompleteness Theorems The Diagonal Slash Argument this objection, he can do so only by adopting rather extraordinary even the set of possible mathematical insights, is also enumerable positions in philosophy of mathematics and in psychology. in the required sense. The last step in Penrose’s argument can Gödel’s incompleteness theorems make use of George Cantor’s The key to Gödel’s theorems and hence Penrose’s arguments is then be taken in one of two ways. If one insists that a thinker (or so-called “Diagonal Slash” argument. Turing used similar the so called “diagonal slash” argument. Let there be a list that any computing agent) really does have this insight which is procedures to show that the Halting Problem was un-decidable. Possible Approaches enumerates all possible real numbers up to infinity. This list might forbidden to him, we simply have an epistemic version of the liar Both of these (independently) cuts the ground off Hilbert’s dream look something like this: A process that can be specified precisely by rules defines a paradox. An alternative conclusion, however, is that the act of of placing “mathematics on a sound logical foundation using the computable function. Adopting Church’s thesis, we can identify the understanding the Turing proof may not in itself constitute having method of formal systems, i.e., finitistic proofs from an agreed 0 . 1 2 3 8 7 3 5 5 …… computable functions with the inputs and outputs of programs on a mathematical proof of the un-provable sentence. upon set of axioms”. 0 . 8 6 3 9 5 2 4 2 …… Turing machines. Other formalisms could be used; all that matters Gödel's first incompleteness theorem, perhaps the single most 0 . 5 3 1 6 4 2 9 9 …… for the arguments here is that it is possible to list the set of all At this point I have not figured out a formal way to go about celebrated result in mathematical logic, states that: 0 . 1 4 3 7 5 8 3 4 …… programs in a natural way, assigning to each of them a so-called proving these two points, but I think they are plausible and could For any consistent formal, recursively enumerable theory that 0 . 9 7 3 0 6 5 7 1 …… Gödel number, or index, based on its place in the list. This be done. The first point seems to be provable using Kleene’s proves basic arithmetical truths, an arithmetical statement that is 0 . 5 3 4 6 1 2 9 7 …… enables us to use numerals to refer to programs; we will call it the Recursion Theorem, and I’m confident of developing a more true, but not provable in the theory, can be constructed. That is, 0 . 1 2 7 8 3 4 5 6 …… standard enumeration. We will refer to the result of running a formal version of the argument. any effectively generated theory capable of expressing elementary 0 . 4 5 3 7 0 9 3 5 …… particular program on a particular input by using the Greek letter arithmetic cannot be both consistent and complete. Φ Φ . , so that (e, x) denotes the result (if it exists) of running Φ References . program e (in the standard enumeration) on input x. is then a Gödel's second incompleteness theorem can be stated as Φ ↑ Φ 1. P. Benacermf, God, the Devil. and Giidel, The Monist 5 I (I . partial function on the integers. We write (e, x) (“ (e, x) follows: 967) 9-32. converges”) to denote that program number e outputs a value, or Φ ↓ Φ 2. G. Boolos, and others, An open peer commentary on The For any formal recursively enumerable (i.e. effectively generated) . halts, on input x, and (e, x) (“ (e, x) diverges”) to mean that theory T including basic arithmetical truths and also certain truths the program fails to terminate when given x as input. Emperor’s New Mind, Behavioral and Brain Sciences 13 (4) about formal provability, T includes a statement of its own (1990) 655.
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