Potential versus actual infinity: insights from reverse mathematics

Steve Simpson (Pennsylvania State University)

Annual Logic Lecture

Every year in the Spring term, the UConn Logic Group hosts a Scholar of Consequence. The Scholar delivers the Annual Logic Lecture, which is open to the public, and over the course of a few days engages in various working sessions with the members of the Group.

Stephen G. Simpson, Pennsylvania State University.

Potential versus actual infinity: insights from reverse mathematics  

In the philosophy of mathematics, there is a crucial distinction between potential infinity and actual infinity. This distinction gives rise to four contrasting viewpoints: ultrafinitism, finitism, predicativism, and infinitism. I am convinced that of these four, finitism is the most objective. This conviction heightens the importance of Hilbert’s program of finitistic reductionism. Some relevant formal systems are PRA, WKL0, IR, ATR0, and ZFC. Foundational research over several decades has revealed that large parts of contemporary mathematics, including the applicable parts, can be formalized in systems such as WKL0 which are finitistically reducible. This seems to provide a possible outline for an objective justification of much of contemporary mathematics.