This looks very like trying to define away something that sure felt like a paradigm shift to the people in the field. Remember that “paradigm” is a belief held by people, not a property inherent in the universe.
Perhaps this is a limitation of my understanding of Kuhn, in that I’m misusing his terminology. I am unaware of mathematics abandoning fundamental objects as inherently misguided the way physics abandoned epicycles or impetus. I expect physics will have similar abandonments in the future, but I expect mathematics never will. The difference is a property of the difference between mathematics and empirical facts. This comment makes the argument I’m trying to assert in slightly different form.
Isn’t that exactly what happened? The phrase “set of all sets that do not contain themselves” isn’t really expressible in Zermelo-Fraenkel set theory, since that has a more limited selection of ways to construct new sets and “the set of everything that satisfies property X” is not one of them.
The “non-expressible in the new concept-space” thing that you think never actually happens.
This looks very like trying to define away something that sure felt like a paradigm shift to the people in the field. Remember that “paradigm” is a belief held by people, not a property inherent in the universe.
Perhaps this is a limitation of my understanding of Kuhn, in that I’m misusing his terminology. I am unaware of mathematics abandoning fundamental objects as inherently misguided the way physics abandoned epicycles or impetus. I expect physics will have similar abandonments in the future, but I expect mathematics never will. The difference is a property of the difference between mathematics and empirical facts. This comment makes the argument I’m trying to assert in slightly different form.
Isn’t that exactly what happened? The phrase “set of all sets that do not contain themselves” isn’t really expressible in Zermelo-Fraenkel set theory, since that has a more limited selection of ways to construct new sets and “the set of everything that satisfies property X” is not one of them.
I don’t think it’s terribly useful to frame the discussion in terms of concepts that never actually happen :-)