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As I understand it, it is possible to prove the consistency of a given axiomatic system using a stronger axiomatic system, but no system can be proven to be absolutely consistent (essentially, the consistency of the given axiomatic system is contingent upon the consistency of the stronger axiomatic system; the system is consistent iff the stronger system is consistent).

Is there a way to conclusively prove the inconsistency of an axiomatic system, apart from simply chancing upon a contradiction as Russell did?

Side notes:

  • I'm assuming the answer is No, since if it were possible to determine if a system is inconsistent, it would be possible to prove if a given statement is decidable in an axiomatic system (this can be done by constructing two modified axiomatic systems - one in which the given statement is true, and another where it is false - and checking which one in inconsistent)
  • A related, slightly naive, question: Is it possible to prove the inconsistency of a stronger axiomatic system by detecting a contradiction in a weaker axiomatic system?
Art
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    What do you mean with "simply chancing upon a contradiction as Russell did?" Russell's discovery of the contradiction implied by Frege's system was a brilliant discovery. – Mauro ALLEGRANZA Nov 10 '19 at 09:30
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    Deriving a contradiction is a proof of inconsistency, "absolute" one. If "stronger axiomatic system" means that it derives all the theorems of "weaker axiomatic system", and the latter derives a contradiction then obviously the former is also inconsistent. If you are asking if there is an algorithm for testing consistency of a given axiomatic system then no, there isn't. – Conifold Nov 10 '19 at 09:42
  • @Conifold Thank you, this answers the question. Is this a consequence of Gödel's theorems? – Art Nov 10 '19 at 10:48
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    It has more to do with the undecidability of the halting problem, of which Gödel's theorems are also corollaries. But it is not straightforward, see Decidability of the consistency for complete finitely axiomatized theories? – Conifold Nov 10 '19 at 22:33

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If a system is inconsistent (which can be shown by deriving a single contradiction), then of course every system containing this system is inconsistent either.

But a stronger system can be inconsistent even if the weaker system is consistent. Just add a contradictionary axiom as $\ 0=1\ $ to the system.

I do not think that we could show that , for example , PA is inconsistent without detecting a contradiction.

Peter
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