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XiXiDu comments on Learning the foundations of math - Less Wrong
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What if Current Foundations of Mathematics are Inconsistent? (Video and discussion.)
At the risk of performing necromancy, I'd like to point out that the video was highly contested afterward on the IAS Foundations of Mathematics mailing list. Many people thought the talk wasn't a reasonable explanation of the current state of affairs in FOM.
I'm not an expert in this, however.
This video turned out to be very interesting (to my naive eyes anyway). It also gives a nice description of formal theories.