David Fernández-Duque - The fractal Goodstein principle

This lecture was part of the Workshop on "Reverse Mathematics: New Paradigms" held at the ESI August 4 - 8, 2025.
Goodstein's theorem is arguably the oldest purely number-theoretic result known to be independent of PA, and is based on representing the natural numbers in terms of the exponential. In recent years, various new Goodstein-like principles of higher proof-theoretic strength have been developed, mostly based on representation systems for natural numbers based on faster-growing functions, such as the Ackermann function. Most recently, variants based on Hardy functions and other fast-growing functions have been shown to be independent of theories of inductive definitions.

In this talk, we develop an alternative Goodstein process, based only on the exponential function, but now using hierarchies of bases to write natural numbers. Thus a number n may be written in some base b, with its "digits" in turn being written in other bases b' smaller b. We will show that this new approach leads to a Goodstein principle of Bachmann-Howard strength, while completely avoiding the use of non-elementary functions in the notation system for natural numbers.

Видео David Fernández-Duque - The fractal Goodstein principle канала Erwin Schrödinger International Institute for Mathematics and Physics (ESI)