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 what do you find different in the two definitions and what do theories have to do with anything j