For a century mathematicians worked to put all of mathematics on an axiomatic basis, where every truth follows from a handful of stated assumptions. A doctoral student in Vienna had already proved the of the first order functional calculus in 1929. Two years later he published something else.
No, and the proof is constructive. In any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of that system, and in particular the of the axioms cannot be proved. That ended a hundred years of attempts to put the whole of mathematics on an axiomatic basis.
Gödel completed his doctoral dissertation under Hahn in 1929 with a thesis proving the completeness of the first order functional calculus, and joined the faculty of the University of Vienna in 1930. The programme he was working inside aimed to put all of mathematics on an axiomatic basis.
In 1931 he published Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, which in English is On formally undecidable propositions of Principia Mathematica and related systems.
The results are about as such. In any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system, and in particular the consistency of the axioms cannot be proved. A hundred years of attempts ended there.
The reading that survives is not that collapsed. The theorem did not destroy its fundamental idea, and it did show that any system would have to be more comprehensive than Hilbert envisaged. The results were a landmark of twentieth-century mathematics, showing that mathematics is not a finished object, and they imply no computer can be programmed to answer every mathematical question.
wat. Every axiomatic system holds statements it can neither prove nor disprove, including its own consistency.
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It changed what mathematics could hope for, and the account of it still used says mathematics is not a finished object.
How do we know?
Gödel published the results in 1931 as On formally undecidable propositions of Principia Mathematica and related systems. He submitted the paper for his habilitation at the University of Vienna, and Hahn accepted it on 1 December 1932.
What is still unknown
The theorem did not destroy the fundamental idea of formalism; it showed that any system would have to be more comprehensive than what Hilbert envisaged.
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