# Talk by Kentaro Fujimoto (LTCS)

**Venue:**

Room A097, ExWi, Sidlerstrasse 5, 3012 Bern

**Speaker:**

Dr. Kentaro Fujimoto (University of Bristol)

**Title:**

Predicates, Predicativity, and Second-order Arithmetic

**Abstract:**

The term "predicativity" alludes to the term "predicates". Quine, however, wrote in his textbook of set theory that they must be firmly dissociated. My first question concerns the relationship of these two notions, predicativity and predicates.

The limit of predicative arithmetic was first characterized in terms of ramified and unramified second-order systems of arithmetic. Nowadays, there are many different types of predicative systems, such as subsystems of set theory, applicative theories, theories of truth, etc. My second question is what the primary significance of second-order arithmetic is in the foundation of mathematics. This question led me to the suggestion that second-order objects should be interpreted as predicates of natural numbers in the context of predicative foundation of mathematics.

The new concept and research of metapredicativity arose in 1990's. Since I first came to know about the research of metapredicativity, I have been wondering what distinguishes predicative, metapredicative, and impredicative arithmetic. My second third question is whether we can extend a justification of predicative arithmetic to metapredicative arithmetic, and, if we can, how far we can extend it. Then, my fourth, more fundamental, question is whether we can draw any conceptually, foundationally, or philosophially rigid distinction between these three.

In this talk, although I have not yet had definite answers, I would like to present and test some ideas toward the answers of these questions. The talk will be more philosophical than mathematical.

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