Past, Present, and Future Directions in Foundations of Mathematics

VideoLectures.Net View the talk in context: http://videolectures.net/godelfellowship2011_friedman_ppf/ View the complete Conference honoring the Winners of the Kurt Gödel Research Prize Fellowships 2008 and 2011: http://videolectures.net/godelfellowship2011_vienna/ Speaker: Harvey Friedman, Department of Mathematics, Ohio State University License: Creative Commons CC BY-NC-ND 3.0 More information at http://videolectures.net/site/about/ More talks at http://videolectures.net/ 1:16 Agenda 2:13 The foundational life 4:40 Foundational life 6:35 Philosophical life 9:10 Mathematical life, scientific life 10:20 An ambition 11:50 Foundations of mathematics foundations of physical science 12:45 Foundations of applied mathematics 14:33 Profound uneasiness - 1 16:13 Profound uneasiness - 2 17:01 First main step epsilon delta 17:33 Number systems 18:38 Real number system multiple definitions 20:17 Infinite upper shift Kernel theorem 21:05 Rationals and integers 22:30 Grand unification 23:59 Pure set theory immutable objects 25:41 Exploding universe 27:21 Assertions and proofs 28:07 Logical structure 29:27 Proof assistants 30:40 Understanding trivialities 32:03 Strict reverse mathematics 33:45 Fundamental foundation moves: consistency, completeness, incompleteness 34:32 Incosistent systems useful? 36:10 Gödel's second incompleteness Theorem 37:04 Four completeness theorems 38:09 Gödel's first incompleteness theorem 39:16 First mathematically natural incompleteness 40:03 Consistency, and the incorporation of new notions 42:25 Concept calculus 44:33 Completeness, and delicate choice of fragments 45:21 0 - Minimality - 1 45:21 0 - Minimality - 2 47:47 Incompleteness, and concreteness, simplicity, naturalness 48:59 Maximal clique embedding