Perhaps, formal system can have proofs for all true statements {omega consistency, completeness}. However, the incompleteness theorem demonstrates that consistent formal system has at least one true statement that has no proof and so is incomplete, as shown by Gödel. Incompleteness theorem shows that the proposition that formal system is omega consistent is not provable by the formal system. For example, logic and number theory have no proof that they are consistent using formal number theory. Therefore, information derivable from formal systems has limits. No axiom set is sufficient to prove all arithmetic or mathematics, by Gödel's proof.
Mathematical Sciences>Mathematics>Axiomatic Theory>Completeness
3-Mathematics-Axiomatic Theory-Completeness
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Date Modified: 2022.0224