According to the link you provided above G del s definition is as follows a class of formulas is not consistent iff there exists a class sign a such that To decipher the notation a class sign is defined as a formula with one free variable of the natural number sort Conseq is the deductive closure of number n is the G del number of the same for the other functions below the numeral for n subst x v y is the result of substitution of y for free occurences of v in x not x is the negation of x and forall x y is the generalization of y by the variable x In usual modern notation this becomes For all n and If we further write T for P for a and for Thm to dispense with the need for G del numbers we get For all n and As far as I can see this is exactly the definition we have in the article modulo the fact that variable v is implicitly restricted to the natural number sort in G del s usage we have instead of the equivalent and we write the formula in words So could you elaborate 1 what do you find different in the two definitions and 2 what do theories have to do with anything J