Comment by hughw

From E.T. Jaynes' Probability Theory [1] :

  To understand the above Gödel result, the essential point is
  the principle of elementary logic that a contradiction
  Ā A implies all propositions, true and false. 
  (Given any two propositions A and B, we have A ⇒ (A+B),
  therefore  Ā A ⇒  Ā (A+B) =  Ā A +  Ā B ⇒ B.) 
  Then let A = {A1, A2, ..., An,} be the system of axioms
  underlying a mathematical theory and T any proposition,
  or theorem, deducible from them:
  
  A ⇒ T.

  Now, whatever T may assert, the fact that T can be deduced
  from the axioms cannot prove that there is no contradiction
  in them, since, if there were a contradiction, T could
  certainly be deduced from them!
  
  This is the essence of the Gödel theorem, as it pertains to
  our problems. As noted by Fisher(1956), it shows us the
  intuitive reason why Gödel's result is true. We do not
  suppose that any logician would accept Fisher's simple
  argument as a proof of the full Gödel theorem; yet for most
  of us it is more convincing than Gödel's long and
  complicated proof.

[1] https://bayes.wustl.edu/etj/prob/book.pdf