Joel David Hamkins: evaluation

Transcript context

…Yeah, that’s great I think. Because Cantor poses the question in the late 19th century. And then it’s totally open. Hilbert asks about it, you know, at the turn of the 20th century. Nobody has any clue. There’s no answer coming. Until 1938, this is four decades later, right? So a long time, and Godel, Kurt Godel proved half of it. What he proved is that if the axioms of set theory are consistent, then there is a set theoretic world where both the axiom of choice and the continuum hypothesis are true. So what he’s doing is showing, this is called the constructible universe, Godel’s L. So he solved this… this is the same result where he answers the safety question of the axiom of choice, but also for the continuum hypothesis. They’re true in the same set theoretic universe we get. So if ZF, without the axiom of choice, is consistent, then so is ZFC plus the continuum hypothesis is the result, 1938. It’s such a beautiful argument. It’s just incredible, I think, because he’s building an alternative mathematical reality. That’s the structure of the proof is that, okay, if there’s any mathematical reality, if there’s any set theoretic world, then we’re going to build another one, a separate one, a different one, maybe different. Maybe it’s the same as the original one. It could be. If we started already in the one that he built, then it would be the same. But there’s no reason to assume it was the same. So he has this kind of model construction method to build this alternative set theoretic reality, the constructible universe, and then he proves that the axiom of choice is true there and also the continuum hypothesis is true there, and it’s just amazing. Really beautiful argument. Okay, so then for the other part of the independence, that’s only half of it, because Godel shows basically that you can’t refute the continuum hypothesis, but that’s not the same thing as proving that it’s true. He showed that if set theory is consistent without the continuing hypothesis, then it’s consistent with the continuing hypothesis. So that’s not the same thing as proving that it’s true. Yeah. And then it didn’t come until 1963 when Paul Cohen invented the method of forcing and proved that if there’s a model of set theory, then there’s a model of set theory in which the continuum hypothesis is false. So Cohen also is giving us this extremely powerful tool for building alternative mathematical realities, is how I think about it. He’s explained to us how to take any set theoretic world and build another different one in which the continuum hypothesis is false. The forcing extension. It’s such a fascinating technique, a tool of forcing. Maybe I’m anthropomorphizing it, but it seems like a way to escape one mathematical universe into another, or to expand it or to alter it. So you travel between mathematical universes. Can you explain the technique of forcing?…

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