Like$\mathbb{N}$s a point of view on natural numbers, II

TL;DR

This paper explores the mathematical structure called likens, focusing on the natural numbers and their properties under addition and multiplication, and characterizes the liken of positive natural numbers within this framework.

Contribution

It introduces a new framework for likens, parameterizes them in an infinite-dimensional space, and characterizes the liken of positive natural numbers using an isomorphism-invariant theorem.

Findings

Liken set can be parameterized by points in an infinite-dimensional metric space.

Main theorem characterizes the liken of positive natural numbers within this space.

Provides a new perspective on natural numbers through the concept of likens.

Abstract

In this paper we continue our research on the concept of liken. This notion has been defined as a sequence of non-negative real numbers, tending to infinity and closed with respect to addition in $\mathbb{R}$. The most important examples of likens are clearly the set of natural numbers $\mathbb{N}$ with addition and the set of positive natural numbers $\mathbb{N}^{*}$ with multiplication, represented by a sequence $(\ln({n+1}))_0^{\infty}$. The set of all likens can be parameterized by the points of some infinite dimensional, complete metric space. In this space of likens we consider elements up to isomorphism and define properties of likens as such, that are isomorphism invariant. The main result of this work is a theorem characterizing the liken ${\mathbb N}^*$ of natural numbers with multiplication in the space of all likens.