:::::According to the link you provided above, Gödel's definition is as follows: a class of formulas κ is not ω-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(κ) is the deductive closure of κ, number(n) is (the Gödel 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 κ, P for a, and  for Thm to dispense with the need for Gödel 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 Gödel'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.