Re:1931: Kurt Gödel shows limits of math, logic, computing, AI
by bidirectional on Thursday June 17, 2021 @10:37AM
(#27540027)
It's not Peano arithmetic, it is basically 'enough of' arithmetic for Goedel's methods to apply. Robinson arithmetic is weaker than PA but Goedel still applies. Goedel's argument is basically a meta-argument about any mathematical system which is rich enough to describe useful mathematics, it does not rely on any particular axiomatisation, rather it applies to all axiomatisations with a few simple features.
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Re:1931: Kurt Gödel shows limits of math, logic, computing, AI
by rssoconnor on Thursday June 17, 2021 @05:49PM
(#27544825)
While it is true that Goedel's theorem applies to weak systems such as Robinson Arithmetic (and any decidable extensions there of), The proof of Goedel's result itself requires at least some amount of induction.
As a consequence the minimum system that Goedel's second incompleteness applies to is stronger than the minimum system that the first incompleteness theorem applies to.
Re:1931: Kurt Gödel shows limits of math, logic, computing, AI
by erichahn on Thursday June 17, 2021 @11:30AM
(#27540679)
I think if Gödels proof needs Q then that is OK.
Q cannot prove its own consistency. Which means there is no way of telling that Gödels theorems are proved in a theory that is inconsistent (where everything is true).