## Link https://youtu.be/fdERLrw9bkY ## Clips ### Reverse Mathematics has gone far beyond the original foundational concerns of the founders of reverse mathematics Friedman and Simpson start = 0:00 end = 1:13 ### It starts with Hilbert's 1900 problem list, specifically problems 1, 2, and 10 start = 1:13 end = 3:22 ### Hilbert's 24th problem was discovered in an unpublished manuscript, and Simpson says to him it sounds like Reverse Mathematics start = 3:22 end = 3:50 ### Hilbert's Second Problem start = 3:50 end = 4:50 ### Some philosophers or would-be philosophers of mathematics believe that Gödel's incompleteness theorem shows the futility of the axiomatic method in mathematics, but we all know the axiomatic method is essential in mathematics, so I want to refute this start = 4:50 end = 5:20 ### The Gödel Hierarchy - For two theories T¹ and T², we say T¹ < T² if the consistency of T¹ is a theorem of T² - Often coincides with the interpretability ordering or the inclusion ordering but not always - The Gödel Hierarchy is (basically) linear and exhibit other regularities start = 5:00 end = 8:00 ### ZFC is extremely strong, medium strength are Z2 down to ACA0, weakest are WKL0 down to bounded arithmetic start = 8:00 end = 10:30 ### What is Foundations of Mathematics? start = 10:15 end = 12:00 ### There are alternative views aside from ZFC. The motivation for Reverse Mathematics is that we want to look at axioms that are appropriate in the sense that they're just barely strong enough to prove the theorems that we want to prove. And it's been found that Second Order Arithmetic and its subsystems are a nice framework for studying this. It's first order arithmetic plus set variables, and within subsystems of second order arithmetic we can formalize large amounts of real mathematics start = 12:00 end = 12:40 ### Jeff Hirst mentioned start = 13:15 end = 13:20 ### This subject goes back to Hilbert and Bernays who looked at Second Order Arithmetic, but they didn't look at Subsystems start = 14:00 end = 16:20 ### Second printing of Subsystems of Second Order Arithmetic because the publisher of the first edition didn't anticipate the great demand for Reverse Mathematics start = 16:20 end = 16:45 ### The Big Five Subsystems of Second Order Arithmetic - RCA0 is formalized computable mathematics, then WKL0, ACA0, ATR0, Π¹₁-CA0 start = 16:45 end = 18:40