What Gödel showed is that, for any consistent, recursively axiomatized formal system, F, strong enough for arithmetic, there are truths expressible in purely arithmetical language which are not provable in F.
In general, for consistent theories T, T ′ containing a sufficiently strong fragment of arithmetic, theory T ′ is of higher consistency strength than theory T just in case T ′ proves that T is consistent.
Note the stress here on 'significant parts'.[35] We know from Gödel's second incompleteness theorem that any consistent and sufficiently strong theory of arithmetic is unable to prove or refute (the formalized statement of) its own consistency.
It should be emphasized that already when it comes to arithmetic, stronger principles are needed in addition to inversion.
Gödel showed that arithmetic has strong expressive powers.
Although these results apply only to mathematical theories strong enough to embed arithmetic, the centrality of the natural numbers (and their extensions into the rationals, reals, complexes, etc).
It is well known that by Gödel's Second Incompleteness Theorem, no consistent, sufficiently strong theory T of arithmetic can prove its own consistency-statement ConT.
A second consequence is that, since provability is weakly representable in any consistent and sufficiently strong formal system for Arithmetic, the general result gives a sentence asserting its own unprovability.
Kurt Gödel's incompleteness theorem demonstrated that any system that is strong enough to express arithmetic is also strong enough to express a formal counterpart of the self-referential proposition in the surprise test example 'This statement cannot be proved in this system'.
Thus there is a φ such that d is provably equivalent to φ, i.e., φ says of itself that it has the property P. A first consequence is that truth cannot be weakly representable in any consistent and sufficiently strong formal system for Arithmetic.
