Hilbert Problems Lesson 3

The Hilbert Problems is a course that explains, one by one, the 23 famous problems that David Hilbert presented in Paris in 1900, at a level suitable for final-year high school and first-year university students. This is Lesson 3: Problem 2: Can Arithmetic Prove Itself Safe? Hilbert's second problem asks for a proof that the axioms of arithmetic can never lead to a contradiction. This lesson explains why one contradiction would ruin everything (the principle of explosion), how consistency is usually shown with models, the Peano axioms and induction, and Hilbert's program of the 1920s. We then follow Kurt Goedel's two great ideas: turning formulas into numbers, and building a sentence that says "I am not provable". This leads to the two incompleteness theorems of 1931, which show that Hilbert's program cannot succeed in its original form. Finally, we see how Gerhard Gentzen (1936) proved the consistency of Peano arithmetic using ordinals up to epsilon zero, and meet Goodstein's theorem, a natural true statement that Peano arithmetic cannot prove (Kirby and Paris, 1982). LESSON SUMMARY 0:00 Introduction 0:10 The question: are the axioms of arithmetic consistent? 1:12 Why one contradiction ruins everything 2:12 Consistency by building a model 3:24 The axioms of arithmetic: Peano and induction 4:34 The program of Hilbert 5:45 Turning formulas into numbers: Goedel numbering 7:09 A sentence about itself 8:26 The first incompleteness theorem 9:25 The second incompleteness theorem 10:44 Gentzen: ordinals and a consistency proof 12:39 A true statement that PA cannot prove: Goodstein's theorem 14:04 The verdict 15:12 Summary Next lesson: Problem 3: cutting solids into pieces and putting them back together (Dehn, 1900). #Hilbert #HilbertProblems #Mathematics #Godel #Incompleteness #Logic #Gentzen #MathEducation