(related to Proposition: Addition Of Natural Numbers)
Let \(n\) and \(m\) be natural numbers and let \(n^+\) denote the successor of \(n\). We will show that the sum \(n+m\) defined recursively by \[\begin{array}{lccl} 1.&n+0&:=&n,\\ 2.&n+m^+&:=&(n+m)^+ \end{array}\quad \quad ( * ) \] exists and is unique. We will show that there is at most one such sum (step 1) and that there is at least one such sum (step 2), thus there is exactly one such sum.
We will show that, for a given \(n\), the definition in \( ( * ) \) is not dependent on the sign of addition. To prove it, we will use two different signs "\(+\)" and "\(\oplus\)" and check, whether or not they define "different" addition operations: 1. Assume \(n+0=n\) as well as \(n\oplus 0=n\). 1. Assume \(n+m^+=(n+m)^+\) as well as \(n\oplus m^+=(n\oplus m)^+\).
Let \(M\) be a subset of all natural numbers \(m\), for which \(n + m=n\oplus m\). We can conclude that \(M\) is not empty, since it (at least) contains the number \(0\). This is because, by first assumption, \[n+0=n=n\oplus 0.\] Now, if any number \(m\) is contained in \(M\), then, according to the Peano axiom P2, and due to the second assumption, we have \[n + m^+=(n+m)^+=(n\oplus m)^+=n\oplus m^+.\] This means that with any number \(m\) contained in \(M\) also its successor \(m^+\) is contained in \(M\). Because also \(0\) is contained in \(M\), we get by Peano axiom P5, that \(M\) contains all natural numbers (principle of induction). This demonstrates, that for a given \(n\) and all \(m\), we have \[n+m=n\oplus m,\] i.e. the sum defined \( (* )\) does not depend on the addition sign "\(+\)".
Let \(M\) be a subset of all natural numbers \(n\), for which \(n + m\) can be defined like in \( ( * ) \). We can conclude that \(M\) is not empty, since it (at least) contains the number \(0\). This is because, by \(1\) of \( ( * ) \), \[0+0=0,\] by \(2\) of \( ( * )\) \[0+m^+=m^+=(0+m)^+.\] Now, if any number \(n\) is contained in \(M\), then according to the Peano axiom P2, we have \[\begin{array}{lccl} 1.&n^++0&:=&n^+,\\ 2.&n^++m^+&:=&(n^++m)^+ \end{array} \] This means that with any number \(n\) contained in \(M\) also its successor \(n^+\) is contained in \(M\). Because also \(0\) is contained in \(M\), we get by Peano axiom P5, that \(M\) contains all natural numbers (principle of induction). This demonstrates, that there is at least one possible definition of an addition operation of natural numbers like in \( ( * ) \).
Altogether, steps \(1\) and \(2\) show that there is exactly one possible definition of addition operation, or that for any two natural numbers \(n\) and \(m\), their sum \(n+m\) exists and is unique.
