The Reverse Mathematics of the Mountain Pass Theorem - Part II

Explore the reverse mathematical analysis of the Mountain Pass Theorem in this 58-minute conference lecture that demonstrates the equivalence between this fundamental result in variational analysis and the weak König's lemma (WKL) over RCA₀. Delve into the proof that the Mountain Pass Theorem of Ambrosetti and Rabinowitz, which provides necessary conditions for the existence of critical points of differentiable functionals on Hilbert spaces, requires exactly the computational strength of WKL within the reverse mathematics framework. Learn how to develop the necessary analytical tools within WKL to access continuous function spaces from [0,1] into separable Banach spaces, and examine the formalized proofs of key components including the deformation lemma and minimax principle. Discover the required theory of ordinary differential equations and understand why this theorem about the existence of an infimum requires only WKL rather than the stronger ACA₀ system. Follow the reversal proof that uses the contrapositive approach, constructing a smooth function that satisfies all hypotheses of the Mountain Pass Theorem but fails its conclusion by assuming the existence of an infinite binary tree with no path.