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> consistency (because it needs to be able to identify e.g. "1+1=3" as false) that's soundness, not consistency. You can build consistent theories of arithmet

> consistency (because it needs to be able to identify e.g. "1+1=3" as false) that's soundness, not consistency. You can build consistent theories of arithmetic in which 1+1=3 is true. The easiest model for such a theory is a structure with a single element. But that's maybe a bad example, as it's too weak for Gödel's theorems to apply to it. So let's consider instead models of Robinson arithmetic. There are models of Robinson arithmetic, for which the sentence "for all x, Sx != x" is false,[0] even though we clearly want it to be true in the natural numbers. So if we add the axiom "there is an x for which Sx = x" to Robinson arithmetic, we get a theory which is arithmetically unsound, so whatever we derive from it, does not readily apply to Turing machines. Yet, Gödel's theorems apply to this theory just as well (assuming it is consistent, which we can't prove, same as we can't prove it for PA). I'll leave it at that. We're talking at cross-purposes, since we can't even get terminology straight. [0]: https://math.stackexchange.com/questions/1066087/model-of-robinson-arithmetic-but-not-peano-arithmetic https://math.stackexchange.com/questions/1066087/model-of-ro...

lisper 4y ago
> But that's maybe a bad example Gee, ya think? > So let's consider instead models of Robinson arithmetic. OK. > There are models of Robinson arithmetic, for which the sentence "for all x, Sx != x" is false OK. So? > even though we clearly want it to be true in the natural numbers. Who is "we"? > if we add the axiom "there is an x for which Sx = x" to Robinson arithmetic, we get a theory which is arithmetically unsound OK. So? > whatever we derive from it, does not readily apply to Turing machines Why do you think that this does not "readily apply to Turing machines"? Turing machines do not natively do arithmetic. They just write symbols to a tape according to a state transition table. Do you really believe that writing a proof checker for Robinson arithmetic, with or without your new axiom, is different in any salient way than writing a proof checker for PA? (And would you please consider the possibility that I might not be a complete idiot?)
- Tainnor 4y ago
  > (And would you please consider the possibility that I might not be a complete idiot?) I do not believe you're an idiot. As a matter of fact, I think you're probably a quite intelligent individual that knows a lot about many topics. Looking at your resume, I do get a strong sense that you're a much better engineer that I ever will be, for example. I do however think that, when it comes to this topic, you're out of your depth, since you don't seem to be aware of concepts and ideas that are taught in a very introductory course on mathematical logic. I don't even think you have digested many of the points I've made in my last post - or in previous ones. It seems like you're refusing to engage, opting instead for picking individual sentences apart as if they were devoid of context. I encourage you to take some time to actually study Gödel's theorems, with all their nuances and the proper historical, mathematical and logical context. The books by Peter Smith seem like a good start: https://www.logicmatters.net/igt/ https://www.logicmatters.net/igt/ - I have only read the long one (which I've linked to previously above), but I suppose the short one should be good too. (And maybe consider that I'm not quite dumb, either. And that mathematical logic is a topic I know a thing or two about.)
  - lisper 4y ago
    It's not your understanding of mathematical logic that I'm questioning, it's your understanding of the theory of computation. In particular, you didn't answer my question: > Do you really believe that writing a proof checker for Robinson arithmetic, with or without your new axiom, is different in any salient way than writing a proof checker for PA?
    - Tainnor 4y ago
      > Do you really believe that writing a proof checker for Robinson arithmetic, with or without your new axiom, is different in any salient way than writing a proof checker for PA? No, and this has exactly nothing to do with my point.
      - lisper 4y ago
        Then why did you bring up RA at all? You must think there is some salient difference between RA and PA with respect to my proof or (I presume) you would not have brought it up. What is it? For that matter, what is the point you are trying to make? You originally wrote [1]: > A theory Γ is said to be complete whenever for all sentences we have either Γ ⊨ φ or Γ ⊨ ¬φ. If you recall, at the time I thought this was a simple mistake on your part, and that what you really meant was "Γ ⊢ φ or Γ ⊢ ¬φ", not "Γ ⊨ φ or Γ ⊨ ¬φ". But you never conceded this. So I'm going to ask you again, is that really what you meant? Because that is not the sense in which the term "completeness" is used in Goedel's incompleteness theorem. The word "complete" is used there to mean syntactic completeness. To quote Wikipedia [2]: "A set of axioms is (syntactically, or negation-) complete if, for any statement in the axioms' language, that statement or its negation is provable from the axioms (Smith 2007, p. 24). THIS IS THE NOTION RELEVANT for Gödel's first Incompleteness theorem. It is NOT TO BE CONFUSED WITH SEMANTIC COMPLETENESS, which means that the set of axioms proves all the semantic tautologies of the given language." [emphasis added] I'm trying to give you the benefit of the doubt here, but you really do appear to be confused on this point, particularly since you keep talking about soundness, which is a semantic property, not a syntactic one. --- [1] https://news.ycombinator.com/item?id=33250168 https://news.ycombinator.com/item?id=33250168 [2] https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Completeness https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...

lisper 4y ago

> But that's maybe a bad example Gee, ya think? > So let's consider instead models of Robinson arithmetic. OK. > There are models of Robinson arithmetic, for which the sentence "for all x, Sx != x" is false OK. So? > even though we clearly want it to be true in the natural numbers. Who is "we"? > if we add the axiom "there is an x for which Sx = x" to Robinson arithmetic, we get a theory which is arithmetically unsound OK. So? > whatever we derive from it, does not readily apply to Turing machines Why do you think that this does not "readily apply to Turing machines"? Turing machines do not natively do arithmetic. They just write symbols to a tape according to a state transition table. Do you really believe that writing a proof checker for Robinson arithmetic, with or without your new axiom, is different in any salient way than writing a proof checker for PA? (And would you please consider the possibility that I might not be a complete idiot?)

Tainnor 4y ago
> (And would you please consider the possibility that I might not be a complete idiot?) I do not believe you're an idiot. As a matter of fact, I think you're probably a quite intelligent individual that knows a lot about many topics. Looking at your resume, I do get a strong sense that you're a much better engineer that I ever will be, for example. I do however think that, when it comes to this topic, you're out of your depth, since you don't seem to be aware of concepts and ideas that are taught in a very introductory course on mathematical logic. I don't even think you have digested many of the points I've made in my last post - or in previous ones. It seems like you're refusing to engage, opting instead for picking individual sentences apart as if they were devoid of context. I encourage you to take some time to actually study Gödel's theorems, with all their nuances and the proper historical, mathematical and logical context. The books by Peter Smith seem like a good start: https://www.logicmatters.net/igt/ https://www.logicmatters.net/igt/ - I have only read the long one (which I've linked to previously above), but I suppose the short one should be good too. (And maybe consider that I'm not quite dumb, either. And that mathematical logic is a topic I know a thing or two about.)
- lisper 4y ago
  It's not your understanding of mathematical logic that I'm questioning, it's your understanding of the theory of computation. In particular, you didn't answer my question: > Do you really believe that writing a proof checker for Robinson arithmetic, with or without your new axiom, is different in any salient way than writing a proof checker for PA?
  - Tainnor 4y ago
    > Do you really believe that writing a proof checker for Robinson arithmetic, with or without your new axiom, is different in any salient way than writing a proof checker for PA? No, and this has exactly nothing to do with my point.
    - lisper 4y ago
      Then why did you bring up RA at all? You must think there is some salient difference between RA and PA with respect to my proof or (I presume) you would not have brought it up. What is it? For that matter, what is the point you are trying to make? You originally wrote [1]: > A theory Γ is said to be complete whenever for all sentences we have either Γ ⊨ φ or Γ ⊨ ¬φ. If you recall, at the time I thought this was a simple mistake on your part, and that what you really meant was "Γ ⊢ φ or Γ ⊢ ¬φ", not "Γ ⊨ φ or Γ ⊨ ¬φ". But you never conceded this. So I'm going to ask you again, is that really what you meant? Because that is not the sense in which the term "completeness" is used in Goedel's incompleteness theorem. The word "complete" is used there to mean syntactic completeness. To quote Wikipedia [2]: "A set of axioms is (syntactically, or negation-) complete if, for any statement in the axioms' language, that statement or its negation is provable from the axioms (Smith 2007, p. 24). THIS IS THE NOTION RELEVANT for Gödel's first Incompleteness theorem. It is NOT TO BE CONFUSED WITH SEMANTIC COMPLETENESS, which means that the set of axioms proves all the semantic tautologies of the given language." [emphasis added] I'm trying to give you the benefit of the doubt here, but you really do appear to be confused on this point, particularly since you keep talking about soundness, which is a semantic property, not a syntactic one. --- [1] https://news.ycombinator.com/item?id=33250168 https://news.ycombinator.com/item?id=33250168 [2] https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Completeness https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...