What is the mathematical meaning of this statement made by Gödel (see details)?

Question

It appears as Proposition VI, in Gödel's 1931 paper "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I" :

"To every $\omega$-consistent recursive class $\kappa$ of formulae there correspond recursive class-signs $r$, such that neither $v \, Gen \, r$ nor $Neg \, (v \, Gen \, r)$ belongs to $Flg (\kappa)$ (where $v$ is the free variable of $r$ )."

P.S. I know the meaning in plain English : "All consistent axiomatic formulations of number theory include undecidable propositions"

Answer 1

See :

• Jean van Heijenoort, From Frege to Gödel : A Source Book in Mathematical Logic (1967), page 592-on.

$\kappa$ is a set of ("decidable") formulas and $Flg(\kappa)$ is the set of consequnces of $\kappa$, i.e. the set of formulas derivable from the formulas in $\kappa$ plus the (logical) axioms by rules of inference.

$\omega$-consistency is a stronger property than consistency (that, under suitable conditions, can be replaced by simple consistency) [see van Heijenoort, page 596].

$Neg(x)$ is the negation of the formula (whose Gödel number is) $x$ :

$Neg(x)$ is the arithmetical function that sends the Gödel number of a formula to the Gödel number of its negation; in other words, $Neg(\ulcorner A \urcorner) = (\ulcorner \lnot A \urcorner)$.

$xGeny$ is the generaliation of $y$ with respect to the variable $x$ :

$Gen(x,y)$ is the arithmetical function (with two arguments) that sends the Gödel number of a variable $x$ and the Gödel number of a formula $A$ to the Gödel number of its universal closure; in other words, $Gen(\ulcorner x \urcorner, \ulcorner A \urcorner) = (\ulcorner \forall x A \urcorner)$.

In conclusion, under the $\omega$-consistency assumption, there is a (unary) predicate $P$ with Gödel number $r$ such that neither $\forall v P(r)$ nor $\lnot \forall v P(r)$ are derivable from the formulas in $\kappa$.

This means that the set $\kappa$ of formulae is incomplete.

See Gödel's Incompleteness Theorems.

See also this "modern" translation of Gödel's original paper.