Kurt Gödel, answered from the texts and cited to the page.
The incompleteness results — there are two of them, and the distinction matters — grew from an attempt to carry out Hilbert's foundational program. Hilbert sought a complete, consistent formal system for mathematics whose consistency could be established by finitary means from within the system itself. I began in 1930 by working on the consistency problem for analysis, seeking to reduce it to that for arithmetic.1
What I found instead was an obstacle rooted in something analogous to the paradoxes of truth and definability — not the paradoxes themselves, which do not apply to precisely specified formal languages, but a non-paradoxical version that arises when one substitutes provability for truth.2 The first theorem: any formal system S in which a sufficient portion of arithmetic can be developed, and which satisfies minimal consistency conditions, is incomplete.
One can construct an elementary arithmetical statement A such that neither A nor its negation is provable in S. The statement so constructed is in fact true — it expresses its own unprovability in S via a representation of the syntax of S within arithmetic itself.3 More precisely, I constructed a sentence G that can be interpreted as asserting of itself that it is not provable in S; if S is consistent, then G is indeed not provable in S, and if S is ω-consistent (a stronger condition ruling out a certain kind of internal incoherence), then the negation of G is also not provable.4
The second theorem follows by formalizing the proof of the first. Within S itself one can prove the conditional: if S is consistent, then G is not provable. But this means S proves W ⊃ G, where W expresses the consistency of S. Hence if S is consistent, S cannot prove W — it cannot prove its own consistency.5 This result forced a fundamental reconsideration of Hilbert's program: if the body of finitary combinatorial reasoning Hilbert required could all be formalized in a single consistent system S, then the program could not be carried out for S or any stronger consistent system.6
One further point worth stating precisely: the undecidable proposition can be expressed in purely arithmetical form — in fact, as a Diophantine equation preceded by mixed universal and existential quantifiers.7 The reach of incompleteness is not confined to some exotic logical stratum; it touches the integers directly. The philosophical weight of the second theorem took me some time to state with full confidence.
In 1931 I was careful to note that the result did not immediately contradict Hilbert's viewpoint, since it was conceivable that finitary methods existed which could not be expressed in the formalism of P. By the time of the 1938 lecture I had convinced myself that all consistency proofs using methods then recognizable as clearly finitary could readily be formalized in Peano arithmetic, and that there was nothing in sight suggesting finitary methods capable of going beyond arithmetic — let alone analysis.8
The hope of carrying out Hilbert's original program, even for classical arithmetic, was gone. What survives — and this is the positive content — is the recognition that mathematical truth exceeds provability in any specific system. We see, from outside S, that G is true; yet S cannot prove it. Truth and provability are not the same relation, and no formal system captures all of objective mathematics.9
In 1930 he began to pursue Hilbert's program for establishing the consistency of formal axiom systems for mathematics by finitary means... Gödel started by working on the consistency problem for analysis, which he sought to reduce to that for arithmetic.Volume I, p. 6
he realized that analogous non-paradoxical arguments could be carried out by substituting the notion of provability for that of truth.Volume I, p. 6
Any formal system S in which a certain amount of theoretical arithmetic can be developed and which satisfies some minimal consistency conditions is incomplete: one can construct an elementary arithmetical statement A such that neither A nor its negation is provable in S. In fact, the statement so constructed is true, since it expresses its own unprovability in S via a representation of the syntax of S in arithmetic.Volume I, p. 6
Gödel constructed a sentence G that can be interpreted as expressing of itself (as given by its number) that G is not provable in S. The first incompleteness theorem tells us that (i) if S is consistent then indeed G is not provable in S, and (ii) if S is ω-consistent then also the negation ~G is not provable in S.Volume I, p. 18
By formalizing the proof of (i), Gödel arrived at the provability within S of W ⊃ G, where W expresses that S is consistent (widerspruchsfrei). Hence if S is consistent then S does not prove W (the second incompleteness theorem).Volume I, p. 18
if the body of finitary combinatorial reasoning that Hilbert required for execution of his consistency program could all be formally developed in a single consistent system S, then the program could not be carried out for S or any stronger (consistent) system.Volume I, p. 6
the undecidable proposition constructed there can be expressed in purely arithmetical form (in fact, as a Diophantine equation preceded by mixed universal and existential quantifiers).Volume I, p. 18
Gödel had convinced himself in the interim that all consistency proofs using methods that were clearly finitary, as then recognized, could readily be formalized in PA, and that there was nothing in sight that suggested finitary methods would be capable of going beyond arithmetic, and certainly not beyond analysis.Collected Works Vol III - Unpublished Essays and Lectures, p. 61
Gödel distinguishes the system of all true mathematical propositions from that of all demonstrable mathematical propositions, calling these mathematics in the objective and subjective senses, respectively, and claims that it is only objective mathematics that no axiom system can fully comprise.Collected Works Vol III - Unpublished Essays and Lectures, pp. 312–313