Gödel’s Incompleteness Theorem

摘要: In this, our final chapter, we shift center of interest first to the foundations mathematics. Section 8.1, shall give a brief historical review formalist approach mathematics and see how Godel’s incompleteness theorem invalidated much Formalist program. 8.2, discuss some general properties recursive logics, yielding proof Incompleteness Theorem. We also follow Myhill’s surprising result that can effectively remove this incompleteness, although never totally but only part at time. To deepen understanding work, present his Completeness Theorem in 8.3, show way which completeness coexist. 8.4 studies speed-up theorems: showing adding an axiom logic may not enable proofs new theorems, dramatic shortening were already available. Finally, 8.5, return main theme book by discussing philosophical controversy centering around implications for question: Are brains essentially superior machines?