# Fractals and the monadic second order theory of one successor

## TL;DR

This paper demonstrates that most classical fractals in real space encode the monadic second-order theory of natural numbers within certain logical structures, revealing deep logical complexity of fractals.

## Contribution

It establishes that a wide class of fractals, including the Cantor set and Sierpinski carpet, interpret the monadic second-order theory of natural numbers in their associated structures.

## Key findings

- Classical fractals interpret the monadic second-order theory of (N,+1).
- Interpretation occurs in structures with real numbers, order, addition, and fractals.
- Results are sharp, matching known interpretations for standard fractals.

## Abstract

We show that if $X$ is virtually any classical fractal subset of $\mathbb{R}^n$, then $(\mathbb{R},<,+,X)$ interprets the monadic second-order theory of $(\mathbb{N},+1)$. This result is sharp in the sense that the standard model of the monadic second-order theory of $(\mathbb{N},+1)$ is known to interpret $(\mathbb{R},<,+,X)$ for various classical fractals $X$ including the middle-thirds Cantor set and the Sierpinski carpet. Let $X \subseteq \mathbb{R}^n$ be closed and nonempty. We show that if the $C^k$-smooth points of $X$ are not dense in $X$ for some $k \geq 1$, then $(\mathbb{R},<,+,X)$ interprets the monadic second-order theory of $(\mathbb{N},+1)$. The same conclusion holds if the packing dimension of $X$ is strictly greater than the topological dimension of $X$ and $X$ has no affine points.