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> This is not at all what it's about. First of all, "incompleteness" has a very specific meaning in logic, namely that the following conditional (what we call "

> This is not at all what it's about. First of all, "incompleteness" has a very specific meaning in logic, namely that the following conditional (what we call "completeness") does not hold hold for some formal system S: > Γ ⊨ φ → Γ ⊢ φ That is, unfortunately, a misunderstanding because the words "(in)completeness" have two very distinct meanings that are both used in logic. One is the completeness of some system of logic, namely what you mentioned: "Γ ⊨ φ → Γ ⊢ φ". That's provably true for fist-order logic (though not for second-order logic), giving rise to Gödel's completeness proof. It remains true when Γ = PA (more specifically, first-order PA). In other words, any truth about numbers that follows from PA has a proof. When we talk about the incompleteness theorems, however, we're talking about the completeness of theories. A theory Γ is said to be complete whenever for all sentences we have either Γ ⊨ φ or Γ ⊨ ¬φ. While there are examples of complete theories (I linked the Wiki article in a comment more up-thread, but it includes simple examples like the theory of dense linear orders sets with endpoints), it is also easy enough to construct a theory that is not complete. In particular (and this follows from the first incompleteness theorem), first-order PA is not complete. What this also implies is that not every true statement about the natural numbers follows from PA, or equivalently, that there are models of PA, i.e. structures in which all axioms of (again, first-order) PA are true, that are nevertheless distinct from the natural numbers because some sentences that are true in the one structure are false in the other. More precisely, the first incompleteness theorem states something like "any effectively axiomatisable theory that is sufficiently strong is incomplete", where "effectively axiomatisable" means that we can determine in finite time whether something is an axiom of that theory, and "sufficiently strong" means something like "can encode every primitive recursive function" (the exact formulation depends a bit, because there are several different "flavours" of the first incompleteness theorem). In particular, it also states that any theory that includes Robinson arithmetic (e.g. PA) is incomplete.

lisper 4y ago
> A theory Γ is said to be complete whenever for all sentences we have either Γ ⊨ φ or Γ ⊨ ¬φ. You mean Γ ⊢ φ or Γ ⊢ ¬φ . But note that what you said is not mutually exclusive with what /u/Tainnor said, i.e. Γ ⊨ φ → Γ ⊢ φ The proof of Godel's theorem is to exhibit a φ for which "Γ ⊢ φ or Γ ⊢ ¬φ" is false. From that you conclude that "Γ ⊨ φ → Γ ⊢ φ" is false on the assumption that "Γ ⊨ φ" is true. But that last part is an informal argument. We believe that "Γ ⊨ G" is true for the Godel sentence G but we (obviously) can't prove it. Indeed, adding ¬G as an axiom to PA produces a consistent system where "Γ ⊢ G" is (trivially) true. The semantics of this system are quite interesting. The semantics of that system say that there exists a Godel number of a proof of G, but no natural number is that number. It is similar to adding axioms like "there exists a number whose successor is zero" or "there exists a number whose square root is the number whose successor is zero", which have obviously been very fruitful extensions of PA.
- Tainnor 4y ago
  > You mean Γ ⊢ φ or Γ ⊢ ¬φ . In principle no, though it's possible that it's defined like that in certain places, since in first-order logic, that amounts to the same thing (given the completeness theorem). But "completeness" is a notion from model theory that doesn't care about proof systems. In a system where the completeness theorem is not true, such as second-order logic with standard semantics, we do have complete theories (such as second-order PA), where we always have Γ ⊨ φ or Γ ⊨ ¬φ, but that doesn't mean that every statement is either provable or disprovable. > The proof of Godel's theorem is to exhibit a φ for which "Γ ⊢ φ or Γ ⊢ ¬φ" is false. From that you conclude that "Γ ⊨ φ → Γ ⊢ φ" is false on the assumption that "Γ ⊨ φ" is true. But that last part is an informal argument. We believe that "Γ ⊨ G" is true for the Godel sentence G but we (obviously) can't prove it. I think that has it a bit backwards. It's not that we believe that PA ⊨ G - in fact, if we believed that, it would immediately imply PA ⊢ G, when that's exactly what we showed to be impossible. It's that we believe that G is true in the natural numbers, which is not the same thing as being true in PA, and that belief rests in our belief that PA is sound (which we can't prove, but seems reasonable). (There are some different "flavours" of the incompleteness theorems, some of them rely on soundness of PA and others only on it being consistent.) > But note that what you said is not mutually exclusive with what /u/Tainnor said, i.e. That's because that's me. ;)
  - lisper 4y ago
    > That's because that's me. ;) Oh, sorry, I thought I was responding to the parent comment by /u/dvt Let's go back to basics. To begin with, let's make sure we're on the same page with regards to notation. "Γ ⊢ φ" means that the string φ is a theorem under Γ. This is a syntactic property of Γ. The claim that some sequence of strings P is a valid proof of φ (and that therefore Γ ⊢ φ is indeed true) can be verified mechanically. "Γ ⊨ φ" means that the string φ is true under some model of Γ. This is a semantic property of Γ and it cannot be checked mechanically. We want "Γ ⊨ φ → Γ ⊢ φ" to be true, that is, we would like it to be the case that for every statement φ that is true under some model of Γ, we can mechanically prove that φ is a theorem of Γ, i.e. we can show that there exists a P which is a proof of φ. If we have such a P in hand we can always check it, but if we don't have a P the question of whether or not one exists is generally an open one. But we actually want more than that. We also want Γ to be "interesting" in some sense. There is no sport in coming up with uninteresting systems for which "Γ ⊨ φ → Γ ⊢ φ" is true. The obvious example is the empty system and a corresponding empty model where every φ is both false and not a theorem. So we impose some minimal structure on Γ, not because the rules of formal systems demand it, but because systems where, say, P and ¬P are both false are not very interesting. So we demand that e.g. there exist at least one true statement and at least one false statement in our model, not because there is any cosmic rule that requires this, but simply because the whole enterprise becomes a pointless exercise in navel-gazing if we don't. That raises the natural question: what is the minimal amount of structure we can impose on Γ to make it "interesting" and worthy of study. And there are many answers to this, because there are many possible things that are potentially "interesting". Propositional logic. Set theory. Yada yada yada. The surprise is that it turns out that with only a very small amount of structure we lose "Γ ⊨ φ → Γ ⊢ φ". As with the imposition of structure in general in order to make systems interesting and worthy of study, there are many different ways in which to impose enough structure to lose this desirable property. One way, the way which historically was the first to be discovered, is to impose enough structure that the system can be modeled by the natural numbers. But this is not the only way. Another way is to impose enough structure that the system can be modeled by a simple machine consisting of a tape and a lookup table. Yet another way is to impose a structure that the system can be modeled by two urns that contain stones, and the ability to add and remove stones from the urns, and look in an urn to see if it is empty or not. There are many many variations on the theme. All of these things turn out to be in some sense "equivalent", and because of that the details of the structure that we impose on the system are uninteresting. It doesn't matter if we use numbers or tapes or stones, there is a certain point beyond which we lose "Γ ⊨ φ → Γ ⊢ φ". And the big surprise is that the threshold is very, very low. It takes very little structure to cross the threshold. There are systems for which "Γ ⊨ φ → Γ ⊢ φ" does hold, but those are necessarily of very limited utility. The proof of the falseness of "Γ ⊨ φ → Γ ⊢ φ" for any given system depends on the details of the system, but the general procedure is always the same: you produce a mapping from proofs to elements of the model. You then use that mapping to produce an element of the model (called G) which is true under the model if and only if there does not exist an element of the model corresponding to a proof of G under your mapping. The key is the mapping from proofs to elements of the model, and Godel numbers were the first such map, but they are not the only such map, and there is nothing particularly special about them other than that they were the first to be discovered. They might have some advantages if you are actually writing out a detailed formal proof of Godel's theorem by hand, just as buggy whips can be useful if you are driving a horse and buggy. But there aren't many compelling reasons to do either one in today's world except as an exercise in nostalgia.

lisper 4y ago

> A theory Γ is said to be complete whenever for all sentences we have either Γ ⊨ φ or Γ ⊨ ¬φ. You mean Γ ⊢ φ or Γ ⊢ ¬φ . But note that what you said is not mutually exclusive with what /u/Tainnor said, i.e. Γ ⊨ φ → Γ ⊢ φ The proof of Godel's theorem is to exhibit a φ for which "Γ ⊢ φ or Γ ⊢ ¬φ" is false. From that you conclude that "Γ ⊨ φ → Γ ⊢ φ" is false on the assumption that "Γ ⊨ φ" is true. But that last part is an informal argument. We believe that "Γ ⊨ G" is true for the Godel sentence G but we (obviously) can't prove it. Indeed, adding ¬G as an axiom to PA produces a consistent system where "Γ ⊢ G" is (trivially) true. The semantics of this system are quite interesting. The semantics of that system say that there exists a Godel number of a proof of G, but no natural number is that number. It is similar to adding axioms like "there exists a number whose successor is zero" or "there exists a number whose square root is the number whose successor is zero", which have obviously been very fruitful extensions of PA.

Tainnor 4y ago
> You mean Γ ⊢ φ or Γ ⊢ ¬φ . In principle no, though it's possible that it's defined like that in certain places, since in first-order logic, that amounts to the same thing (given the completeness theorem). But "completeness" is a notion from model theory that doesn't care about proof systems. In a system where the completeness theorem is not true, such as second-order logic with standard semantics, we do have complete theories (such as second-order PA), where we always have Γ ⊨ φ or Γ ⊨ ¬φ, but that doesn't mean that every statement is either provable or disprovable. > The proof of Godel's theorem is to exhibit a φ for which "Γ ⊢ φ or Γ ⊢ ¬φ" is false. From that you conclude that "Γ ⊨ φ → Γ ⊢ φ" is false on the assumption that "Γ ⊨ φ" is true. But that last part is an informal argument. We believe that "Γ ⊨ G" is true for the Godel sentence G but we (obviously) can't prove it. I think that has it a bit backwards. It's not that we believe that PA ⊨ G - in fact, if we believed that, it would immediately imply PA ⊢ G, when that's exactly what we showed to be impossible. It's that we believe that G is true in the natural numbers, which is not the same thing as being true in PA, and that belief rests in our belief that PA is sound (which we can't prove, but seems reasonable). (There are some different "flavours" of the incompleteness theorems, some of them rely on soundness of PA and others only on it being consistent.) > But note that what you said is not mutually exclusive with what /u/Tainnor said, i.e. That's because that's me. ;)
- lisper 4y ago
  > That's because that's me. ;) Oh, sorry, I thought I was responding to the parent comment by /u/dvt Let's go back to basics. To begin with, let's make sure we're on the same page with regards to notation. "Γ ⊢ φ" means that the string φ is a theorem under Γ. This is a syntactic property of Γ. The claim that some sequence of strings P is a valid proof of φ (and that therefore Γ ⊢ φ is indeed true) can be verified mechanically. "Γ ⊨ φ" means that the string φ is true under some model of Γ. This is a semantic property of Γ and it cannot be checked mechanically. We want "Γ ⊨ φ → Γ ⊢ φ" to be true, that is, we would like it to be the case that for every statement φ that is true under some model of Γ, we can mechanically prove that φ is a theorem of Γ, i.e. we can show that there exists a P which is a proof of φ. If we have such a P in hand we can always check it, but if we don't have a P the question of whether or not one exists is generally an open one. But we actually want more than that. We also want Γ to be "interesting" in some sense. There is no sport in coming up with uninteresting systems for which "Γ ⊨ φ → Γ ⊢ φ" is true. The obvious example is the empty system and a corresponding empty model where every φ is both false and not a theorem. So we impose some minimal structure on Γ, not because the rules of formal systems demand it, but because systems where, say, P and ¬P are both false are not very interesting. So we demand that e.g. there exist at least one true statement and at least one false statement in our model, not because there is any cosmic rule that requires this, but simply because the whole enterprise becomes a pointless exercise in navel-gazing if we don't. That raises the natural question: what is the minimal amount of structure we can impose on Γ to make it "interesting" and worthy of study. And there are many answers to this, because there are many possible things that are potentially "interesting". Propositional logic. Set theory. Yada yada yada. The surprise is that it turns out that with only a very small amount of structure we lose "Γ ⊨ φ → Γ ⊢ φ". As with the imposition of structure in general in order to make systems interesting and worthy of study, there are many different ways in which to impose enough structure to lose this desirable property. One way, the way which historically was the first to be discovered, is to impose enough structure that the system can be modeled by the natural numbers. But this is not the only way. Another way is to impose enough structure that the system can be modeled by a simple machine consisting of a tape and a lookup table. Yet another way is to impose a structure that the system can be modeled by two urns that contain stones, and the ability to add and remove stones from the urns, and look in an urn to see if it is empty or not. There are many many variations on the theme. All of these things turn out to be in some sense "equivalent", and because of that the details of the structure that we impose on the system are uninteresting. It doesn't matter if we use numbers or tapes or stones, there is a certain point beyond which we lose "Γ ⊨ φ → Γ ⊢ φ". And the big surprise is that the threshold is very, very low. It takes very little structure to cross the threshold. There are systems for which "Γ ⊨ φ → Γ ⊢ φ" does hold, but those are necessarily of very limited utility. The proof of the falseness of "Γ ⊨ φ → Γ ⊢ φ" for any given system depends on the details of the system, but the general procedure is always the same: you produce a mapping from proofs to elements of the model. You then use that mapping to produce an element of the model (called G) which is true under the model if and only if there does not exist an element of the model corresponding to a proof of G under your mapping. The key is the mapping from proofs to elements of the model, and Godel numbers were the first such map, but they are not the only such map, and there is nothing particularly special about them other than that they were the first to be discovered. They might have some advantages if you are actually writing out a detailed formal proof of Godel's theorem by hand, just as buggy whips can be useful if you are driving a horse and buggy. But there aren't many compelling reasons to do either one in today's world except as an exercise in nostalgia.