Monday, August 1, 2011

Ultimate logic: To infinity and beyond - New Scientist - New Scientist

Ultimate logic: To infinity and beyond: At the same time, though, Gödel had a crazy-sounding hunch about how you might fill in most of these cracks in mathematics' underlying logical structure: you simply build more levels of infinity on top of it. That goes against anything we might think of as a sound building code, yet Gödel's guess turned out to be inspired. He proved his point in 1938. By starting from a simple conception of sets compatible with Zermelo and Fraenkel's rules and then carefully tailoring its infinite superstructure, he created a mathematical environment in which both the axiom of choice and the continuum hypothesis are simultaneously true. He dubbed his new world the "constructible universe" - or simply "L"...

As Cohen was able to show, in some logically possible worlds the hypothesis is true and there is no intermediate level of infinity between the countable and the continuum; in others, there is one; in still others, there are infinitely many. With mathematical logic as we know it, there is simply no way of finding out which sort of world we occupy...

Among other things, ultimate L provides for the first time a definitive account of the spectrum of subsets of the real numbers: for every forking point between worlds that Cohen's methods open up, only one possible route is compatible with Woodin's map. In particular it implies Cantor's hypothesis to be true, ruling out anything between countable infinity and the continuum.

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