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Explore the mathematical theory of better-quasi-orderings through ordinal analysis in this 56-minute lecture from the Summer School on "Reverse Mathematics: New Paradigms" at the Erwin Schrödinger International Institute. Delve into the strengthened notion of well-foundedness for non-total orders, focusing on C. Nash-Williams' concept of better-quasi-ordering (bqo) that proved instrumental in R. Laver's proof of Fraïssé's conjecture. Examine how bqo theory requires unusually strong axioms from a reverse mathematics perspective, where even proving that all finite orders are bqo necessitates at least arithmetical recursion along natural numbers (equivalent to omega-th Turing jumps, forming the theory ACA_0^+). Discover how ordinal analysis provides the only known proof method for this result, connecting G. Gentzen's foundational work showing Peano arithmetic's proof-theoretic ordinal epsilon_0 to the analysis of finite bqos. Learn how these two mathematical frameworks converge to establish fundamental results about finite orders, bridging classical proof theory with modern reverse mathematics.
