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Results tagged with foundations
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user 8991
Mathematical logic, Set theory, Peano arithmetic, Model theory, Proof theory, Recursion theory, Computability theory, Univalent foundations, Reverse mathematics, Frege foundation of arithmetic, Goedel's incompleteness and Mathematics, Structural set theory, Category theory, Type theory.
6
votes
What is the proof-theoretic ordinal of Hyperarithmetical Comprehension?
The difficulty is in what it means to go up the ramified hierarchy. When talking about theories, you can't write down a computable or c.e. theory which perfectly captures the whole ramified hierarchy …
- 7,145
5
votes
What is the proof-theoretic ordinal of Hyperarithmetical Comprehension?
This is a second answer (think of it as part two of the other, already long, answer), in response to the clarification of the question by Keshav Srinivasan in the comments on that question.
First, no …
- 7,145
15
votes
Consistency of Analysis (second order arithmetic)
As Noah says, the direct successor of Gentzen's method, cut-elimination, has been generalized up to $\Pi^1_2$-comprehension. This was shown separately by Rathjen and Arai; the full results have never …
- 7,145
10
votes
Does formalizing math require search and creativity, or is it near-mechanical?
I'm not an expert (I've played around with HOL a few times, but never fully formalized an interesting theorem), but my experience was that it's similar to writing up a technical proof after you've wor …
76
votes
Accepted
How should a "working mathematician" think about sets? (ZFC, category theory, urelements)
(Note that this is a feature of ZFC, not of set theoretic foundations in general. Quine's New Foundations allows certain self-containing sets.) …
- 7,145