# Representing Ordinal Numbers with Arithmetically Interesting Sets of Real Numbers

## TL;DR

This paper explores how real numbers and specific sets of natural numbers can be used to represent various ordinal numbers through modular multiplication, revealing limitations and possibilities depending on set restrictions.

## Contribution

It introduces a framework linking real numbers, sets of natural numbers, and order types, highlighting the impact of set restrictions on representability of ordinal structures.

## Key findings

- Irrational x can generate any order type with suitable A.
- If A is thin, only certain order types are possible.
- Restricting A to powers of 2 limits the order types achievable.

## Abstract

For a real number $x$ and set of natural numbers $A$, define $x \ast A := \{ x a \bmod 1: a\in A\}\subseteq [0,1).$ We consider relationships between $x$, $A$, and the order-type of $x\ast A$. For example, for every irrational $x$ and order-type $\alpha$, there is an $A$ with $x\ast A \simeq \alpha$, but if $\alpha$ is a well order, then $A$ must be a thin set. If, however, $A$ is restricted to be a subset of the powers of 2, then not every order type is possible, although arbitrarily large countable well orders arise.