4 ms·

Is the problem with finding the Gödel sentence, or with proving it? I believe the issue is the proving it. For a sentence to be a Gödel sentence of something,

by drdeca 7y ago

Is the problem with finding the Gödel sentence, or with proving it? I believe the issue is the proving it. For a sentence to be a Gödel sentence of something, that something, I think, should be a formal system, a set of inference rules and axioms. While there may generally not be much issue in talking about "the Gödel sentence of [a process/thing that uses such a system]" to refer to the Gödel sentence of the system that the thing uses, in this case, I think there is no issue. Finding the Gödel sentence of a system is, iirc, a fairly straightforward computational process, given a good description of the system. So, it should be no difficulty to have a program that takes its definition of a system, and checks proofs in that system / looks for proofs in that system, and also computes the Gödel sentence of that system. Furthermore, I think (but am slightly less sure, though still fairly sure) that consistent and sound systems can prove things like "if the system [description of self goes here] can prove [Gödel sentence of the system], then it is not consistent." . Furthermore, I see nothing establishing that humans can do any better than computers in this regard. Be careful when using the Gödel incompleteness theorems as justifications of positions! They are often misinterpreted.

yters 7y ago
Humans perhaps cannot goedelize themselves, or perhaps it's not possible to formalize the mind. However, in either case, it also intuitively seems that humans can goedelize any finite, formal system. Thus, the human mind is not reducible to a finite, formal system.
- drdeca 7y ago
  A human only can fit only finitely much information in their memory. There are formal systems that no human could fit a description of into their mind. Also, the way humans reason seems more like something paraconsistent than something truly self-consistent. A person can be convinced of something by some argument, and then convinced the other way. It is true that the human mind is not an example of [thing which the incompleteness theorems show cannot exist], but this is not surprising.
  - yters 7y ago
    We can augment our memory with physical storage. We might not analyze within the lifetime or capacity of the universe (or multiverse), but that's a practical limitation, not a theoretical one. Humans are irrational and all, but we make surprisingly consistent formal systems. I don't think an inconsistent system could do this, so this suggests the human mind is something beyond finite formal systems. For example, we could model this ability as access to an infinite table of all possible Turing machines and their halting status, perhaps with some noise added to represent our irrationality. We would still have something beyond the limitations of goedel's theorem, as well as computation and randomness, which would account for our intuition in the matter. So there is no good reason to dismiss the idea there is something special about the human mind not captured by formal systems.
    - drdeca 7y ago
      I do agree that there is something special about the human mind not captured by formal systems (in the "set of axioms or axiom schemas + inference rules" sense). (not so much for empirical external evidence reasons, so much as for like, what I want to call "teleological reasons"? Like, my goals and values are more coherent/less nonsensical assuming that there is something special, so it makes sense (as far as my goals are concerned) for me to assume that there is something special.) However, I'm not so sure that this something special includes anything Gödel related. As to whether an "inconsistent system" could "make" consistent formal systems, I'm not entirely sure what you mean. A formal system (in the sense of set of axioms + inference rules) is not an agent, so I'm not sure what you mean about it "making" things. We talk about a formal system prov"ing" something, but perhaps a more clear way of saying what we mean by that, is that the system admits a proof of the thing, or that there is a proof of the thing, for/within that system. If by a system "making" a system, we just mean that it "proves" (as in, "admits a proof") that some other system is consistent, then I see no reason that a paraconsistent system couldn't show that some other system is consistent. Have you seen the "Logical Induction" paper by MIRI ? If a system is inconsistent, then there is a proof that it is inconsistent. Logical induction assigns "probabilities" to different mathematical statements, and it can be shown that these probabilities converge in reasonable ways. If there is a proof of a statement in the system that the logical induction thing is working under, then the "probability" of that statement will converge to 1. Also, this statements that logical induction applies probabilities to include statements about itself. I think that, if some particular system is self consistent, then, well, for one thing, the probability for the statement "that system is inconsistent" will not go to 1, and furthermore, that there are statements of "the probability for 'that system is inconsistent' will be below c at step t" (for t much larger than the current time step) which will (I think) go to 1. In that sense, I think logical induction could probably do a good job at concluding that some systems are probably self consistent. Now, I'm not totally sure of that. I don't 100% understand the logical induction results. But, yeah. Seems relevant to me.

by drdeca 7y ago

yters 7y ago

Humans perhaps cannot goedelize themselves, or perhaps it's not possible to formalize the mind. However, in either case, it also intuitively seems that humans can goedelize any finite, formal system. Thus, the human mind is not reducible to a finite, formal system.

drdeca 7y ago
A human only can fit only finitely much information in their memory. There are formal systems that no human could fit a description of into their mind. Also, the way humans reason seems more like something paraconsistent than something truly self-consistent. A person can be convinced of something by some argument, and then convinced the other way. It is true that the human mind is not an example of [thing which the incompleteness theorems show cannot exist], but this is not surprising.
- yters 7y ago
  We can augment our memory with physical storage. We might not analyze within the lifetime or capacity of the universe (or multiverse), but that's a practical limitation, not a theoretical one. Humans are irrational and all, but we make surprisingly consistent formal systems. I don't think an inconsistent system could do this, so this suggests the human mind is something beyond finite formal systems. For example, we could model this ability as access to an infinite table of all possible Turing machines and their halting status, perhaps with some noise added to represent our irrationality. We would still have something beyond the limitations of goedel's theorem, as well as computation and randomness, which would account for our intuition in the matter. So there is no good reason to dismiss the idea there is something special about the human mind not captured by formal systems.
  - drdeca 7y ago
    I do agree that there is something special about the human mind not captured by formal systems (in the "set of axioms or axiom schemas + inference rules" sense). (not so much for empirical external evidence reasons, so much as for like, what I want to call "teleological reasons"? Like, my goals and values are more coherent/less nonsensical assuming that there is something special, so it makes sense (as far as my goals are concerned) for me to assume that there is something special.) However, I'm not so sure that this something special includes anything Gödel related. As to whether an "inconsistent system" could "make" consistent formal systems, I'm not entirely sure what you mean. A formal system (in the sense of set of axioms + inference rules) is not an agent, so I'm not sure what you mean about it "making" things. We talk about a formal system prov"ing" something, but perhaps a more clear way of saying what we mean by that, is that the system admits a proof of the thing, or that there is a proof of the thing, for/within that system. If by a system "making" a system, we just mean that it "proves" (as in, "admits a proof") that some other system is consistent, then I see no reason that a paraconsistent system couldn't show that some other system is consistent. Have you seen the "Logical Induction" paper by MIRI ? If a system is inconsistent, then there is a proof that it is inconsistent. Logical induction assigns "probabilities" to different mathematical statements, and it can be shown that these probabilities converge in reasonable ways. If there is a proof of a statement in the system that the logical induction thing is working under, then the "probability" of that statement will converge to 1. Also, this statements that logical induction applies probabilities to include statements about itself. I think that, if some particular system is self consistent, then, well, for one thing, the probability for the statement "that system is inconsistent" will not go to 1, and furthermore, that there are statements of "the probability for 'that system is inconsistent' will be below c at step t" (for t much larger than the current time step) which will (I think) go to 1. In that sense, I think logical induction could probably do a good job at concluding that some systems are probably self consistent. Now, I'm not totally sure of that. I don't 100% understand the logical induction results. But, yeah. Seems relevant to me.