THE LOGIC OF INFINITE: GEORG CANTOR AND HIERARCHY INFINITY

A schoolmaster's son from Saint Petersburg, stranded for life in a provincial German university where nobody important ever visited, sat down in 1873 to answer a question that had been considered improper for two thousand years. How many numbers are there? Not many. Not endlessly many. How many, exactly. The answer Georg Cantor found broke the question in half. Some infinities, he proved, are strictly larger than others, and the proof fits on a postcard. It also fits nowhere in the mathematics of his century. Colleagues told him the work was a disease. A cardinal in Rome read his letters with more sympathy than his own department showed him. He founded the German Mathematical Society mainly because no existing society would have him, spent years arguing that Francis Bacon wrote Shakespeare, endured repeated hospitalizations, and died in a clinic in Halle during a wartime famine, three years before David Hilbert announced that nobody would drive anyone out of the paradise Cantor had built. Boris Kriger and Anna Kriger tell that life in full, and alongside it the stranger story of what Cantor's argument turned out to be. The diagonal is not a trick for comparing sizes. It is a general-purpose machine for stepping out of any system that claims to contain everything, and once you have seen it work, you find it everywhere: in Gödel's undecidable sentence, in Turing's halting problem, in Russell's paradox, in the question of whether the physical universe has room for a single one of Cantor's infinities. Which, this book argues, it does not. Along the way: what Aristotle actually banned and why, why a honeymoon was spent talking mathematics, how a theological argument about God's exclusivity nearly settled a question in set theory, what Kurt Gödel and Paul Cohen did to the tower of infinities, and why the hierarchy of the title is not a ladder anyone climbs. Keywords: Georg Cantor, set theory, infinity, mathematics history, philosophy of mathematics, transfinite numbers, continuum hypothesis