Traditionally, logic was restricted to proofs having a finite number of steps and to expressions of finite length. Around 1954–56, infinitely long formulas entered the mainstream of mathematical logic through the work of Henkin, Karp, Scott, and Tarski. Soon Hanf and Tarski used such logics to settle negatively the 30-year-old problem of whether the first strongly inaccessible cardinal is measurable, a result Tarski communicated to the first LMPS congress in 1960. Infinitary logic continues to be fertile in unexpected ways, as shown by Kolaitis at the present congress.