Natural Number Addition is Commutative

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Theorem

The operation of addition on the set of natural numbers $\N$ is commutative:

$\forall m, n \in \N: m + n = n + m$


Proof 1

Consider the natural numbers defined as a naturally ordered semigroup.


By definition, the operation in a naturally ordered semigroup is commutative.

Hence the result.

$\blacksquare$


Proof 2

Proof by induction.

Consider the natural numbers $\N$ defined as the elements of the minimal infinite successor set $\omega$.


From the definition of addition in $\omega$‎, we have that:

\(\displaystyle \forall m, n \in \N: \ \ \) \(\displaystyle m + 0\) \(=\) \(\displaystyle m\)
\(\displaystyle m + n^+\) \(=\) \(\displaystyle \left({m + n}\right)^+\)


For all $n \in \N$, let $P \left({n}\right)$ be the proposition:

$\forall m \in \N: m + n = n + m$


Basis for the Induction

From Natural Number Addition Commutes with Zero, we have:

$\forall m \in \N: m + 0 = m = 0 + m$

Thus $P \left({0}\right)$ is seen to be true.


This is our basis for the induction.


Induction Hypothesis

Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 0$, then it logically follows that $P \left({k^+}\right)$ is true.


So this is our induction hypothesis $P \left({k}\right)$:

$\forall m \in \N: m + k = k + m$


Then we need to show that $P \left({k^+}\right)$ follows directly from $P \left({k}\right)$:

$\forall m \in \N: m + k^+ = k^+ + m$


Induction Step

This is our induction step:


\(\displaystyle k^+ + m\) \(=\) \(\displaystyle \left({k + m}\right)^+\) Natural Number Addition Commutativity with Successor
\(\displaystyle \) \(=\) \(\displaystyle \left({m + k}\right)^+\) from the induction hypothesis
\(\displaystyle \) \(=\) \(\displaystyle m + k^+\) by definition

So $P \left({k}\right) \implies P \left({k^+}\right)$ and the result follows by the Principle of Mathematical Induction.


Therefore:

$\forall m, n \in \N: m + n = n + m$

$\blacksquare$


Proof 3

In the axiomatisation of $1$-based natural numbers, this is rendered:

$\forall x, y \in \N_{> 0}: x + y = y + x$


Using the following axioms:

\((A)\)   $:$     \(\displaystyle \exists_1 1 \in \N_{> 0}:\) \(\displaystyle a \times 1 = a = 1 \times a \)             
\((B)\)   $:$     \(\displaystyle \forall a, b \in \N_{> 0}:\) \(\displaystyle a \times \paren {b + 1} = \paren {a \times b} + a \)             
\((C)\)   $:$     \(\displaystyle \forall a, b \in \N_{> 0}:\) \(\displaystyle a + \paren {b + 1} = \paren {a + b} + 1 \)             
\((D)\)   $:$     \(\displaystyle \forall a \in \N_{> 0}, a \ne 1:\) \(\displaystyle \exists_1 b \in \N_{> 0}: a = b + 1 \)             
\((E)\)   $:$     \(\displaystyle \forall a, b \in \N_{> 0}:\) \(\displaystyle \)Exactly one of these three holds:\( \)             
\(\displaystyle a = b \lor \paren {\exists x \in \N_{> 0}: a + x = b} \lor \paren {\exists y \in \N_{> 0}: a = b + y} \)             
\((F)\)   $:$     \(\displaystyle \forall A \subseteq \N_{> 0}:\) \(\displaystyle \paren {1 \in A \land \paren {z \in A \implies z + 1 \in A} } \implies A = \N_{> 0} \)             


Let $x \in \N_{> 0}$ be arbitrary.

For all $n \in \N_{> 0}$, let $P \left({n}\right)$ be the proposition:

$x + n = n + x$


Basis for the Induction

From Natural Number Commutes with 1 under Addition, we have that:

$\forall x \in \N_{> 0}: x + 1 = 1 + x$

and so $P \left({1}\right)$ holds.

This is our basis for the induction.


Induction Hypothesis

Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 1$, then it logically follows that $P \left({k + 1}\right)$ is true.


So this is our induction hypothesis:

$x + k = k + x$


Then we need to show:

$x + \left({k + 1}\right) = \left({k + 1}\right) + x$


Induction Step

This is our induction step:


\(\displaystyle x + \left({k + 1}\right)\) \(=\) \(\displaystyle \left({x + k}\right) + 1\) Natural Number Addition is Associative
\(\displaystyle \) \(=\) \(\displaystyle \left({k + x}\right) + 1\) Induction Hypothesis
\(\displaystyle \) \(=\) \(\displaystyle k + \left({x + 1}\right)\) Natural Number Addition is Associative
\(\displaystyle \) \(=\) \(\displaystyle k + \left({1 + x}\right)\) Basis for the Induction
\(\displaystyle \) \(=\) \(\displaystyle \left({k + 1}\right) + x\) Natural Number Addition is Associative

So $P \left({k}\right) \implies P \left({k + 1}\right)$ and the result follows by the Principle of Mathematical Induction.

$\blacksquare$


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