Research project Potential infinity in the foundations of mathematics
0, 1, 2, ... These are the natural numbers. We learn to count at the age of four. But should we conceive of this series as infinite? The project employs potential infinity to explain core phenomena in the foundations of mathematics.
Add principles for addition and multplication, logical concepts such as ”for all,” and some basic axioms, and we have the system called arithmetics. This is of fundamental significance for mathematics, and the archtype of infinity. Infinity can be explained in various ways. Some philosophers (ultrafinitists) claim that there is no mathematical infinity, whereas others (actualists) claim that there is a completed set of infinitely many numbers
Already Aristotle advocated for a middle ground between these extremes, which, however, hasn’t received a modern logical articulation until an article by Linnebo and Shapiro in 2018. The idea is to consider 0, 1, 2, ... as a potentially infinite process: whichever number we choose, it is finite, but we can always choose a larger number.
Project description
The standard model N of arithmetic contains the numbers 0, 1, 2,... A non-standard model M extends N with larger non-standard numbers, such that M still behaves much like N. Intuitively, the new numbers are infinite from the outside, but finite from the inside of M. This duality makes non-standard models a powerful tool, enabling a whole scientific domain of fundamental importance to logic. Nonetheless, there is a distinct lack of positive explanations of non-standard models in the philosophical literature.
The PI has invented a technique, combining the philosphical methods of indexical reference with recent developments on potential infinity (the idea that N is not a completed whole, but rather a process of indefinitely extendible finite structures) to explain the positive role played by non-standard numbers. The PI also has a new potentialist interpretation of the proof of the Löwenheim-Skolem theorem, which promises to shed new light on Skolem's paradox
Project members
Project managers
Paul Gorbow
Postdoc
