...theorems.
The subtle point of first showing that the 5#5's exist before defining addition on the natural numbers was overlooked by Peano, and later by Landau who was following Peano (who had defined 6#6 after defining +). Grundjot discovered this flaw in Landau's work, and repairs were made following ideas of Kalmar.
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...theory
Of course Gödel had found a statement in first-order number theory which could not be derived from PA, but it was not the sort of statement one would encounter in traditional number theory.
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Stan Burris
Fri Jan 31 13:26:00 EST 1997