On Formally Undecidable Propositions (1931)
Über formal unentscheidbare Sätze — the incompleteness theorems
1931 paper proving that rich enough formal systems contain true arithmetical statements they cannot prove, if consistent.
Primary publication. Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik, 38, 173–198. DOI: 10.1007/BF01700692
Standard English access. Jean van Heijenoort (ed.), From Frege to Gödel: A Source Book in Mathematical Logic (Harvard University Press, 1967); or Gödel, Collected Works, Vol. I.
Claims we use in Genesis
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First incompleteness theorem. Any consistent formal system capable of expressing basic arithmetic contains statements that are true (in the standard model) but not provable in the system.
→ Tier 0 (Rⁿ) is not identical to “provable in S.” Tier-one formal captures and tier-two theorems remain fallible relative to the Real. -
Second incompleteness theorem. If such a system S is consistent, Con(S) is not provable in S.
→ Meta-mathematical claims about a foundation’s consistency are falsify-only or tier 12+ unless proved in a stronger meta-system.