Sam Sanders - Uniformity in Mathematics

Explore a lecture on uniformity in mathematics, focusing on highly uniform theorems and their impact on logical systems. Delve into historical examples from the 19th century, examining how compactness was used in real analysis proofs. Investigate the development of uniform results through gauge interpretation techniques. Analyze the implications of these highly uniform theorems on Reverse Mathematics and computability theory, challenging traditional hierarchies of logical systems. Examine specific examples such as the intermediate value theorem and Ramsey's theorem. Consider how these uniform approaches may question the distinction between direct and indirect proofs, suggesting that classical 'proof by contradiction' can be more effective and require weaker axioms than elementary direct proofs. Cover topics including the Gödel hierarchy, Cousin's lemma, HBU, integration beyond Riemann and Lebesgue, and the Monty Hall problem of logic.

Intro
Historical examples of uniformity
Uniform theorems, then and now
The Gödel hierarchy
Gödel hierarchy (including higher-order arithmetic)
Cousin's lemma and HBU
Beyond Riemann and Lebesgue integration
Reverse Mathematics of HBU The following properties of the
Known 'uniform' theorems
New uniform theorems: Ramsey's theorem
The Monty Hall problem of logic
Collapsing the Godel hierarchy
Kleene: the marriage of recursion theory and intuitionism