Joel David Hamkins: evaluation

Transcript context

…I think this is a good moment to talk about Gödel’s incompleteness theorems. So, can you explain them and what do they teach us about the nature of mathematical truth? Absolutely. It’s one of the most profound developments in mathematical logic. I mean, the incompleteness theorems are when mathematical logic, in my view, first became sophisticated. It’s a kind of birth of the subject of mathematical logic. But to understand the theorems, you really have to start a little bit earlier with Hilbert’s program because at that time, you know, with the Russell Paradox and so on, there were these various contradictions popping up in various parts of set theory and the Burali-Forti paradox and so on. And Hilbert was famously supportive of set theory. I mean, there’s this quote of him saying, “No one shall cast us from the paradise that Cantor has created for us.” And what I take him to mean by that is he was so captured by the idea of using set theory as a foundation of mathematics, and it was so powerful and convenient and unifying in a way that was extremely important. And he didn’t want to give that up, despite the danger of these paradoxes, these contradictions, basically, is how some people viewed them. And so, it’s… This minefield of paradoxes. Yeah.…

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