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Even though set theory purports to offer a foundation for all of mathematical practice, it would seem very odd to hear someone say, say, "I work in a narrow branch of set theory called partial differential equations."

My question: how would a set theory expert in 2011 characterize the scope, or purview, or principal goals of his or her subject? I have particular interest in aspects of the answer that might differ from what an expert might have said perhaps 20 years ago.

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This is not exactly an answer, but I think that this text might be relevant to your question: – Haim Nov 20 '11 at 6:13
1… may prove to be a useful reading. – Asaf Karagila Nov 20 '11 at 6:49
with respect to the PDE joke, you might enjoy – Michael Blackmon Nov 23 '11 at 15:18

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