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You can have an infinite chain of predecessors of non-standard numbers, each claiming being FINITE(). That satisfies your definition. This could be easily demon

You can have an infinite chain of predecessors of non-standard numbers, each claiming being FINITE(). That satisfies your definition. This could be easily demonstrated in Robinson arithmetic, which has simple, understandable non-standard models.

lisper 2y ago
> You can have an infinite chain of predecessors of non-standard numbers, each claiming being FINITE(). Ah. That makes sense. Thanks. So surely someone has come up with a system that captures the intuitive spirit of mathematical induction, i.e. that the chain has to actually be rooted at 0 rather than have an infinite chain of predecessors or be a loop? How hard could that be?
- btilly 2y ago
  How hard could that be? It seems like it should be straightforward. But it's mathematically impossible. Similarly, it seemed to David Hilbert like it should be possible to have a system of formal logic that all could accept, which would prove itself to be consistent. But that also proved to be impossible. And the logic which lead to that being proved impossible is deeply connected to the first line of reasoning that I gave you for why this is also impossible. And in the end what they all come down to is some kind of self-reference. In this case, to have a finite statement define what it means to be finite, we must already know what finite means. Second order logic sort of assumes that we can simply know what finite means. First order logic doesn't allow you to wave your hands like that. You can't cheat in any way. And so you can't find a way to do it.
  - lisper 2y ago
    > Second order logic sort of assumes that we can simply know what finite means. First order logic doesn't allow you to wave your hands like that. OK, this I do not understand. How does 2ndOL "sort of assume that we can simply know what finite means"? That seems like an extraordinarily imprecise characterization of 2ndOL. 2ndOL is a formal system just like 1stOL. Neither one lets you "wave your hands" about anything AFAICT.
    - btilly 2y ago
      Read https://plato.stanford.edu/entries/logic-higher-order/ https://plato.stanford.edu/entries/logic-higher-order/ if you wish. I view the "for all properties" bit as handwaving. See particularly section 6 of that article, which at one point gives: The semantics of second-order logic depending on these hard questions of set theory puts second-order logic into the same “basket” with set theory. Criticism of set theory becomes criticism of second-order logic and vice versa. My Constructivist tendencies are due to my discomfort with accepting the validity of set-theoretic constructions with decisions based on unverifiable truths. Accepting second order logic requires accepting quantifying over similar collections of unverifiable claims. I'm going to be no more sympathetic.
      - lisper 2y ago
        Thanks.
  - Tainnor 2y ago
    > Similarly, it seemed to David Hilbert like it should be possible to have a system of formal logic that all could accept, which would prove itself to be consistent. Minor nitpick, but I don't think this fully captures the essence of Hilbert's program. If we had a system that could prove itself consistent, that would mean nothing, because the system could be inconsistent, therefore unsound and so its consistency "proof" wouldn't mean anything. Rather, the goal was to start from an arguably "uncontroversial" base of mathematics (something like primitive recursive arithmetic, which intuitionists might also accept) and use that to bootstrap "infinitary mathematics" of the kind the formalists such as Hilbert were arguing for. The hope was that one could prove the consistency of such more abstract mathematics (PA or ZFC) from the more concrete foundations. Unfortunately, Gödel's theorems completely demolish that idea. It's not just that a (sufficiently strong) theory can't prove itself consistent, it also can't be proven consistent by a strictly weaker theory such as PRA (because if such a proof existed in the weaker system, it would of course also exist in original system). Interestingly, there are systems which are neither stronger nor weaker than PA which can prove PA consistent: https://en.m.wikipedia.org/wiki/Gentzen%27s_consistency_proof https://en.m.wikipedia.org/wiki/Gentzen%27s_consistency_proo...
    - btilly 2y ago
      That goal is what I meant by the phrase "that all could accept". You're right that I could have expanded on it though. That said, Gentzen's result is new to me. Though I have seen other results that are somewhat similar.
btilly 2y ago
Robinson arithmetic is an excellent example here. Using ZFC, Robinson can take a standard model of the reals and construct a nonstandard model, and a function from the standard model to the nonstandard model. Every possible statement of first order logic that can be made about the standard model is true in the nonstandard model. Likewise every possible statement of first order logic that can be made in the nonstandard model only involving things that came from the standard model are also true of the standard model. This is his "transfer principle". However the nonstandard model comes with things like infinitesmals, and infinite numbers. So in the nonstandard world we can really do things like define a derivative as dy/dx, or an integral as an infinite sum. Then recover the classical definitions through the transfer principle. See https://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-management/ https://terrytao.wordpress.com/2007/06/25/ultrafilters-nonst... for an explanation of how this actually works in some detail. But if you get your head around it, you'll see exactly why first order logic cannot possibly capture the idea of being a standard model. Because, in first order logic, the nonstandard model really is indistinguishable. And so finite cannot be expressed in first order logic.
- zajio1am 2y ago
  Note that i meant https://en.wikipedia.org/wiki/Robinson_arithmetic https://en.wikipedia.org/wiki/Robinson_arithmetic , while you wrote about results from different Robinson: https://en.wikipedia.org/wiki/Nonstandard_analysis https://en.wikipedia.org/wiki/Nonstandard_analysis .
  - btilly 2y ago
    Oops, thank you for the correction! I learned something today. Though I'm sure you can see how I thought that the other Robinson fitted in the conversation,

lisper 2y ago

> You can have an infinite chain of predecessors of non-standard numbers, each claiming being FINITE(). Ah. That makes sense. Thanks. So surely someone has come up with a system that captures the intuitive spirit of mathematical induction, i.e. that the chain has to actually be rooted at 0 rather than have an infinite chain of predecessors or be a loop? How hard could that be?

btilly 2y ago
How hard could that be? It seems like it should be straightforward. But it's mathematically impossible. Similarly, it seemed to David Hilbert like it should be possible to have a system of formal logic that all could accept, which would prove itself to be consistent. But that also proved to be impossible. And the logic which lead to that being proved impossible is deeply connected to the first line of reasoning that I gave you for why this is also impossible. And in the end what they all come down to is some kind of self-reference. In this case, to have a finite statement define what it means to be finite, we must already know what finite means. Second order logic sort of assumes that we can simply know what finite means. First order logic doesn't allow you to wave your hands like that. You can't cheat in any way. And so you can't find a way to do it.
- lisper 2y ago
  > Second order logic sort of assumes that we can simply know what finite means. First order logic doesn't allow you to wave your hands like that. OK, this I do not understand. How does 2ndOL "sort of assume that we can simply know what finite means"? That seems like an extraordinarily imprecise characterization of 2ndOL. 2ndOL is a formal system just like 1stOL. Neither one lets you "wave your hands" about anything AFAICT.
  - btilly 2y ago
    Read https://plato.stanford.edu/entries/logic-higher-order/ https://plato.stanford.edu/entries/logic-higher-order/ if you wish. I view the "for all properties" bit as handwaving. See particularly section 6 of that article, which at one point gives: The semantics of second-order logic depending on these hard questions of set theory puts second-order logic into the same “basket” with set theory. Criticism of set theory becomes criticism of second-order logic and vice versa. My Constructivist tendencies are due to my discomfort with accepting the validity of set-theoretic constructions with decisions based on unverifiable truths. Accepting second order logic requires accepting quantifying over similar collections of unverifiable claims. I'm going to be no more sympathetic.
    - lisper 2y ago
      Thanks.
- Tainnor 2y ago
  > Similarly, it seemed to David Hilbert like it should be possible to have a system of formal logic that all could accept, which would prove itself to be consistent. Minor nitpick, but I don't think this fully captures the essence of Hilbert's program. If we had a system that could prove itself consistent, that would mean nothing, because the system could be inconsistent, therefore unsound and so its consistency "proof" wouldn't mean anything. Rather, the goal was to start from an arguably "uncontroversial" base of mathematics (something like primitive recursive arithmetic, which intuitionists might also accept) and use that to bootstrap "infinitary mathematics" of the kind the formalists such as Hilbert were arguing for. The hope was that one could prove the consistency of such more abstract mathematics (PA or ZFC) from the more concrete foundations. Unfortunately, Gödel's theorems completely demolish that idea. It's not just that a (sufficiently strong) theory can't prove itself consistent, it also can't be proven consistent by a strictly weaker theory such as PRA (because if such a proof existed in the weaker system, it would of course also exist in original system). Interestingly, there are systems which are neither stronger nor weaker than PA which can prove PA consistent: https://en.m.wikipedia.org/wiki/Gentzen%27s_consistency_proof https://en.m.wikipedia.org/wiki/Gentzen%27s_consistency_proo...
  - btilly 2y ago
    That goal is what I meant by the phrase "that all could accept". You're right that I could have expanded on it though. That said, Gentzen's result is new to me. Though I have seen other results that are somewhat similar.

btilly 2y ago

Robinson arithmetic is an excellent example here. Using ZFC, Robinson can take a standard model of the reals and construct a nonstandard model, and a function from the standard model to the nonstandard model. Every possible statement of first order logic that can be made about the standard model is true in the nonstandard model. Likewise every possible statement of first order logic that can be made in the nonstandard model only involving things that came from the standard model are also true of the standard model. This is his "transfer principle". However the nonstandard model comes with things like infinitesmals, and infinite numbers. So in the nonstandard world we can really do things like define a derivative as dy/dx, or an integral as an infinite sum. Then recover the classical definitions through the transfer principle. See https://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-management/ https://terrytao.wordpress.com/2007/06/25/ultrafilters-nonst... for an explanation of how this actually works in some detail. But if you get your head around it, you'll see exactly why first order logic cannot possibly capture the idea of being a standard model. Because, in first order logic, the nonstandard model really is indistinguishable. And so finite cannot be expressed in first order logic.

zajio1am 2y ago
Note that i meant https://en.wikipedia.org/wiki/Robinson_arithmetic https://en.wikipedia.org/wiki/Robinson_arithmetic , while you wrote about results from different Robinson: https://en.wikipedia.org/wiki/Nonstandard_analysis https://en.wikipedia.org/wiki/Nonstandard_analysis .
- btilly 2y ago
  Oops, thank you for the correction! I learned something today. Though I'm sure you can see how I thought that the other Robinson fitted in the conversation,