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guslFeferman - Does Reductive Proof Theory Have a Viable Rationale?
He starts out saying that reductions in science tend to be explanatory, but in mathematics their purpose is foundational.
He debates against Karl-Georg Niebergall (who I met recently) about the merits of implementing reductions as proof-theoretic reduction vs. as relative theory-interpretations. The latter seems to favor the latter. But I don't see why the two "implementations" have to be incompatible.
I like Feferman's coinage of "set-theoretic imperalism", a dogmatic view.
KG Niebergall - On the Logic of Reducibility: Axioms and Examples (doesn't seem to be available unless you are using it from a subscribing institution: I have asked the author for a copy)
Ed Zalta - Neo-Logicism? An Ontological Reduction of Mathematics to Metaphysics
explains mathematics using his theory of abstract objects
Albert Visser - An Overview of Interpretability Logic
Exactly 2 years ago, I had a mini-course with Visser (in Utrecht) about categories of interpretation. Unfortunately, he was too busy to help me through the steep learning curve.
I was interested in applying the ideas of theory interpretations to formalizations of science, and it seems Niebergall has done exactly this.
---
Btw, what is the possible-world semantics of a provability logic?
If such a semantics is viable, then it holds that:
a formula phi is provable in world w (written w |= [] phi) if and only if phi is true in all worlds that w points to.
What are these possible worlds and what is the relation between them?
He starts out saying that reductions in science tend to be explanatory, but in mathematics their purpose is foundational.
He debates against Karl-Georg Niebergall (who I met recently) about the merits of implementing reductions as proof-theoretic reduction vs. as relative theory-interpretations. The latter seems to favor the latter. But I don't see why the two "implementations" have to be incompatible.
I like Feferman's coinage of "set-theoretic imperalism", a dogmatic view.
KG Niebergall - On the Logic of Reducibility: Axioms and Examples (doesn't seem to be available unless you are using it from a subscribing institution: I have asked the author for a copy)
Ed Zalta - Neo-Logicism? An Ontological Reduction of Mathematics to Metaphysics
explains mathematics using his theory of abstract objects
Albert Visser - An Overview of Interpretability Logic
Exactly 2 years ago, I had a mini-course with Visser (in Utrecht) about categories of interpretation. Unfortunately, he was too busy to help me through the steep learning curve.
I was interested in applying the ideas of theory interpretations to formalizations of science, and it seems Niebergall has done exactly this.
---
Btw, what is the possible-world semantics of a provability logic?
If such a semantics is viable, then it holds that:
a formula phi is provable in world w (written w |= [] phi) if and only if phi is true in all worlds that w points to.
What are these possible worlds and what is the relation between them?