If with an axiomatic system there are undecidable propositions, that is not the same with the axiomatic system being contradictory, i.e. where you can prove that a proposition is both true and false.
An undecidable proposition is neither true nor false, it is not both true and false.
A system with undecidable propositions may be perfectly fine, while a contradictory system is useless.
Thus what the previous poster has said has nothing to do with what Gödel had proved.
Ensuring that the system of axioms that you use is non-contradictory has remained as useful today as by the time of Euclid and basing your reasoning on clearly stated non-contradictory axioms has also remained equally important, even if we are now aware that there may be undecidable things (which are normally irrelevant in practice anyway).
The results of Gödel may be interpreted as a demonstration that the use of ternary logic is unavoidable in mathematics, like it already was in real life, where it cannot always be determined whether a claim is true or false.
Indeed. Soundness and completeness are different things.
There are two well accepted definitions of soundness. One of them is the inability to prove true == false, that is, one cannot prove a contradiction from within that axiomatic system.
If with an axiomatic system there are undecidable propositions, that is not the same with the axiomatic system being contradictory, i.e. where you can prove that a proposition is both true and false.
An undecidable proposition is neither true nor false, it is not both true and false.
A system with undecidable propositions may be perfectly fine, while a contradictory system is useless.
Thus what the previous poster has said has nothing to do with what Gödel had proved.
Ensuring that the system of axioms that you use is non-contradictory has remained as useful today as by the time of Euclid and basing your reasoning on clearly stated non-contradictory axioms has also remained equally important, even if we are now aware that there may be undecidable things (which are normally irrelevant in practice anyway).
The results of Gödel may be interpreted as a demonstration that the use of ternary logic is unavoidable in mathematics, like it already was in real life, where it cannot always be determined whether a claim is true or false.
Indeed. Soundness and completeness are different things.
There are two well accepted definitions of soundness. One of them is the inability to prove true == false, that is, one cannot prove a contradiction from within that axiomatic system.
They aren't completely different, because trying to achieve wine generally harms the other.
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