Contents -- I: FOUNDATIONAL PROBLEMS -- 1 Deciding the undecidable: Wrestling with Hilbert's problems -- 2 Infinity in mathematics: Is Cantor necessary? -- II: FOUNDATIONAL WAYS -- 3 The logic of mathematical discovery versus the logical structure of mathematics -- 4 Foundational ways -- 5 Working foundations -- III: GÖDEL -- 6 Gödel's life and work -- 7 Kurt Gödel: Conviction and caution -- 8 Introductory note to Gödel's 1933 lecture -- IV: PROOF THEORY -- 9 What does logic have to tell us about mathematical proofs? -- 10 What rests on what? The proof-theoretic analysis of mathematics -- 11 Gödel's Dialectica interpretation and its two-way stretch -- V: COUNTABLY REDUCIBLE MATHEMATICS -- 12 Infinity in mathematics: Is Cantor necessary? (Conclusion) -- 13 Weyl vindicated: Das Kontinuum seventy years later -- 14 Why a little bit goes a long way: Logical foundations of scientifically applicable mathematics -- Symbols -- References -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Z.