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In order to understand Gödel’s theorems, one must first explain the key concepts essential to it, such as “formal system”, “consistency”, and “completeness”. Roughly, a formal system is a system ...
#1: Initial revision
by
Julius H.
·
2024-02-20T12:58:08Z (9 months ago)
What is the definition of ‘decidability’?
> In order to understand Gödel’s theorems, one must first explain the key concepts essential to it, such as “formal system”, “consistency”, and “completeness”. Roughly, a formal system is a system of axioms equipped with rules of inference, which allow one to generate new theorems. The set of axioms is required to be finite or at least decidable, i.e., there must be an algorithm (an effective method) which enables one to mechanically decide whether a given statement is an axiom or not. If this condition is satisfied, the theory is called “recursively axiomatizable”, or, simply, “axiomatizable”. The rules of inference (of a formal system) are also effective operations, such that it can always be mechanically decided whether one has a legitimate application of a rule of inference at hand. Consequently, it is also possible to decide for any given finite sequence of formulas, whether it constitutes a genuine derivation, or a proof, in the system—given the axioms and the rules of inference of the system. https://plato.stanford.edu/Entries/goedel-incompleteness/ I believe this quality is called ‘decidability’. How is this property expressed formally, in logical notation?