Question about consistency of axiomatic systems

Question

I often heard that if you can prove from some axiomatic system some proposition "P", and you can prove proposition "not P", then it means that you can prove any proposition in that axiomatic system. But why is it so ? Is it possible to prove this somehow using proof theory or mathematical logic ?

Answer 1

See Ex falso (or Principle of explosion): in many proof systems we have that from $P∧¬P$ we can derive $Q$, for $Q$ whatever.

This rule can be a "basic" rule of the system (like the ($\bot$E) rule of Natural Deduction) or it can be derived from other basic rules (see Wiki's proof).

The rules is grounded in the definition of Logical consequence: a formula $Q$ whatever is logical consequence of an inconsistent set of premises.

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But see Paraconsistent Logic for an overview of systems rejecting Ex falso.