6 ms·

> Moreover, Robinson arithmetic can be interpreted in general set theory, a small fragment of ZFC. https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t

by IsTom 12d ago

> Moreover, Robinson arithmetic can be interpreted in general set theory, a small fragment of ZFC. https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory#Consistency https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...

andriy_koval 12d ago
> interpreted its hard to me to tell what this means formally(as I said I am not expert). There is no "interpret" operator in zfc. I believe what it says if you add some robinson axioms + some logical rules on top of zfc, you can carry your results.
- IsTom 12d ago
  It's the same way you don't need to have GCD in stdlib to say that you can compute GCD in C++. You can make your own using parts given. You don't need to add any axioms, you just build some sets to represent numbers and make operations that act the same way as arithmetic, define some equality relations. Then you derive rules of arithmetic for your handcrafted arithmetic using ZF axioms and you're good. You get axioms of arithmetic derived from your regular axioms without adding them as new axioms to your theory.
  - andriy_koval 11d ago
    > you just build some sets to represent numbers and make operations that act the same way as arithmetic which is already "just" some non trivial problem(there is no "operations" in set theory), and we are discussing if it is achievable.
    - IsTom 11d ago
      You make relations and functions out of sets and prove theorems about them, reducing definition of things in terms of belonging to a set. This isn't particularly complicated.
      - andriy_koval 11d ago
        No, once you start formalize this, it becomes complicated. There is a reason why looks like there is no "peano can be derived from zfc" theorem which would close dispute, and my opponents need to throw links on bro math from stackexchange in this discussion.
        IsTom 11d ago
        Per https://en.wikipedia.org/wiki/Peano_axioms#Set-theoretic_models https://en.wikipedia.org/wiki/Peano_axioms#Set-theoretic_mod... > The Peano axioms can be derived from set theoretic constructions of the natural numbers and axioms of set theory such as ZF.[15] If you're going against the general consensus you should present something more than nebulous assertions that it's wrong.
        andriy_koval 11d ago
        Obviously citation from wikipedia can't be considered as replacement of math proof. > If you're going against the general consensus you should present something more than nebulous assertions that it's wrong. burden of proof is on the one who claims something exists.

by IsTom 12d ago

andriy_koval 12d ago

> interpreted its hard to me to tell what this means formally(as I said I am not expert). There is no "interpret" operator in zfc. I believe what it says if you add some robinson axioms + some logical rules on top of zfc, you can carry your results.

IsTom 12d ago
It's the same way you don't need to have GCD in stdlib to say that you can compute GCD in C++. You can make your own using parts given. You don't need to add any axioms, you just build some sets to represent numbers and make operations that act the same way as arithmetic, define some equality relations. Then you derive rules of arithmetic for your handcrafted arithmetic using ZF axioms and you're good. You get axioms of arithmetic derived from your regular axioms without adding them as new axioms to your theory.
- andriy_koval 11d ago
  > you just build some sets to represent numbers and make operations that act the same way as arithmetic which is already "just" some non trivial problem(there is no "operations" in set theory), and we are discussing if it is achievable.
  - IsTom 11d ago
    You make relations and functions out of sets and prove theorems about them, reducing definition of things in terms of belonging to a set. This isn't particularly complicated.
    - andriy_koval 11d ago
      No, once you start formalize this, it becomes complicated. There is a reason why looks like there is no "peano can be derived from zfc" theorem which would close dispute, and my opponents need to throw links on bro math from stackexchange in this discussion.
      - IsTom 11d ago
        Per https://en.wikipedia.org/wiki/Peano_axioms#Set-theoretic_models https://en.wikipedia.org/wiki/Peano_axioms#Set-theoretic_mod... > The Peano axioms can be derived from set theoretic constructions of the natural numbers and axioms of set theory such as ZF.[15] If you're going against the general consensus you should present something more than nebulous assertions that it's wrong.
        andriy_koval 11d ago
        Obviously citation from wikipedia can't be considered as replacement of math proof. > If you're going against the general consensus you should present something more than nebulous assertions that it's wrong. burden of proof is on the one who claims something exists.