4 ms·

Wittgenstein was one of the first to realize incorrectness of Gödel's famous 1931 article on the inferential undecidability (incompleteness) of Russell's system

Wittgenstein was one of the first to realize incorrectness of Gödel's famous 1931 article on the inferential undecidability (incompleteness) of Russell's system Principia Mathematica. For more information please see the following: https://papers.ssrn.com/abstract=3603021 https://papers.ssrn.com/abstract=3603021 Professionals (including Gödel) ridiculed Wittgenstein's work on logic and ignored his correct important argument against [Gödel 1931].

abhishekjha 6y ago
Can you elaborate a little more on how Wittgenstein showed the incorrectness of incompleteness theorem?
- ProfHewitt 6y ago
  Wittgenstein was one of the first to argue that adding the [Gödel 1931] proposition I’mUnprovable results in contradiction by providing the following correct but somewhat convoluted argument [Wittgenstein 1937-1944] which has been explained by adding explanations of steps in brackets: “Let us suppose [Gödel 1931 was correct and therefore] I prove the unprovability (in Russell’s system) of [Gödel’s proposition] P [that is, ⊢⊬P where P⇔⊬P]; then by this proof I have proved P [⊢P because P⇔⊬P]. Now if this proof were one in Russell’s system [⊢⊢P] — I should in this case have proved at once that it belonged [⊢P] and did not belong [⊢¬P because ¬P⇔⊢P] to Russell’s system. But there is a contradiction here! [⊢P as well as ⊢¬P] …[This] is what comes of making up such propositions.” However, Wittgenstein did not point out that the [Gödel 1931] proposition I’mUnprovable is not allowed by the rules of Russell’s system because of restrictions on orders of propositions. Nothing of practical importance depends on the existence of Gödel’s proposition I’mUnprovable. As discussed in https://papers.ssrn.com/abstract=3603021 https://papers.ssrn.com/abstract=3603021 , the important property of inferential undecidability (incompleteness) of Russell’s system can be proved in a different way without using I’mUnprovable. Furthermore, having Gödel's monster proposition I’mUnprovable comes at the heavy cost of introducing another monster, namely, “A powerful theory cannot prove its own consistency” as discussed here: https://papers.ssrn.com/abstract=3603021 https://papers.ssrn.com/abstract=3603021

abhishekjha 6y ago

Can you elaborate a little more on how Wittgenstein showed the incorrectness of incompleteness theorem?

ProfHewitt 6y ago
Wittgenstein was one of the first to argue that adding the [Gödel 1931] proposition I’mUnprovable results in contradiction by providing the following correct but somewhat convoluted argument [Wittgenstein 1937-1944] which has been explained by adding explanations of steps in brackets: “Let us suppose [Gödel 1931 was correct and therefore] I prove the unprovability (in Russell’s system) of [Gödel’s proposition] P [that is, ⊢⊬P where P⇔⊬P]; then by this proof I have proved P [⊢P because P⇔⊬P]. Now if this proof were one in Russell’s system [⊢⊢P] — I should in this case have proved at once that it belonged [⊢P] and did not belong [⊢¬P because ¬P⇔⊢P] to Russell’s system. But there is a contradiction here! [⊢P as well as ⊢¬P] …[This] is what comes of making up such propositions.” However, Wittgenstein did not point out that the [Gödel 1931] proposition I’mUnprovable is not allowed by the rules of Russell’s system because of restrictions on orders of propositions. Nothing of practical importance depends on the existence of Gödel’s proposition I’mUnprovable. As discussed in https://papers.ssrn.com/abstract=3603021 https://papers.ssrn.com/abstract=3603021 , the important property of inferential undecidability (incompleteness) of Russell’s system can be proved in a different way without using I’mUnprovable. Furthermore, having Gödel's monster proposition I’mUnprovable comes at the heavy cost of introducing another monster, namely, “A powerful theory cannot prove its own consistency” as discussed here: https://papers.ssrn.com/abstract=3603021 https://papers.ssrn.com/abstract=3603021