Ordering on Natural Numbers is Trichotomy

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Theorem

Let $\N$ be the natural numbers.

Let $<$ be the (strict) ordering on $\N$.


Then exactly one of the following is true:

$(1): \quad a = b$
$(2): \quad a > b$
$(3): \quad a < b$

That is, $<$ is a trichotomy on $\N$.


Proof

Applying the definition of $<$, the theorem becomes:


Exactly one of the following is true:

$(1): \quad a = b$
$(2): \quad \exists n \in \N_{>0} : b + n = a$
$(3): \quad \exists n \in \N_{>0} : a + n = b$


We will use the principle of Mathematical Induction.

Let $P \left({a}\right)$ be the proposition that exactly one of the above is true, for all natural numbers $b$, for fixed natural number $a$.


Basis for the Induction

We will prove that the proposition is true for $a = 0$.

Using Proof by Cases, we divide the proposition into two cases.


Case 1: $b = 0$

$(1)$ is true

It follows trivially from the values of $a$ and $b$.


$(2)$ is false

Aiming for a contradiction, suppose $c$ is a non-zero natural number such that:

$b + c = a$

Substituting the values of $a$ and $b$, we obtain:

$0 + c = 0$

By Natural Number Addition Commutes with Zero, we can simplify the equation to:

$c = 0$

which is a contradiction of the assumption that $c$ is non-zero.


$(3)$ is false

Aiming for a contradiction, suppose $c$ is an non-zero natural number such that:

$a + c = b$

Substituting the values of $a$ and $b$, we obtain:

$0 + c = 0$

By Natural Number Addition Commutes with Zero, we can simplify the equation to:

$c = 0$

which is a contradiction of the assumption that $c$ is non-zero.


Case 2: $b \ne 0$.

$(1)$ is false

It follows trivially from the fact that $a = 0$.


$(2)$ is false

Aiming for a contradiction, suppose $c$ is an non-zero natural number such that:

$b + c = a$

Substituting the values of $a$, we obtain:

$b + c = 0$

By Non-Successor Element of Peano Structure is Unique, there exists a natural number $d$ such that:

$s \left({d}\right) = c$

where $s$ denotes the successor mapping.

Therefore, we have:

$b + s \left({d}\right) = 0$

Applying the definition of addition in Peano structure, we get:

$s \left({b + d}\right) = 0$

which is a contradiction of $(P4)$: $0$ is not in the image of $s$.


$(3)$ is true

By Natural Number Addition Commutes with Zero, we have:

$a + b = b$

The result follows.


Inductive hypothesis

It is now assumed that the proposition is true for $a = k$, where $k$ is a natural number.

That is, for all natural numbers $b$, exactly one of the following is true:

$(1): \quad k = b$
$(2): \quad k > b$
$(3): \quad k < b$


Then, it is to be proved that the proposition is true for $a = s \left({k}\right)$.

That is, for all natural numbers $b$, exactly one of the following is true:

$(1'): \quad s \left({k}\right) = b$
$(2'): \quad s \left({k}\right) > b$
$(3'): \quad s \left({k}\right) < b$


Inductive step

We have:

\(\displaystyle s \left({k}\right)\) \(=\) \(\displaystyle s \left({k + 0}\right)\) Definition of Addition in Peano Structure
\(\displaystyle \) \(=\) \(\displaystyle k + s \left({0}\right)\) Definition of Addition in Peano Structure
\(\displaystyle \) \(=\) \(\displaystyle k + 1\) Definition of One


Case 1: $k = b$

In this case:

\(\displaystyle s \left({k}\right)\) \(=\) \(\displaystyle k + 1\)
\(\displaystyle \) \(=\) \(\displaystyle b + 1\)


$(1')$ is false

Aiming for a contradiction, suppose $(1')$ is true.

Then:

\(\displaystyle s \left({k}\right)\) \(=\) \(\displaystyle b\) Assumption
\(\displaystyle b + 1\) \(=\) \(\displaystyle b\)
\(\displaystyle 1\) \(=\) \(\displaystyle 0\) Natural Number Addition is Cancellable
\(\displaystyle s \left({0}\right)\) \(=\) \(\displaystyle 0\) Definition of One

which is a contradiction of $(P4)$: $0$ is not in the image of $s$.


$(2')$ is true

This is apparent from the definition of $>$.


$(3')$ is false

Aiming for a contradiction, suppose $(3')$ is true.

Let $c$ be a non-zero natural number such that:

$s \left({k}\right) + c = b$


Then:

\(\displaystyle s \left({k}\right) + c\) \(=\) \(\displaystyle b\) Assumption
\(\displaystyle \left({b + 1}\right) + c\) \(=\) \(\displaystyle b\)
\(\displaystyle b + \left({c + 1}\right)\) \(=\) \(\displaystyle b\) Natural Number Addition is Associative and Natural Number Addition is Commutative
\(\displaystyle c + 1\) \(=\) \(\displaystyle 0\) Natural Number Addition is Cancellable
\(\displaystyle c + s \left({0}\right)\) \(=\) \(\displaystyle 0\) Definition of One
\(\displaystyle s \left({c}\right)\) \(=\) \(\displaystyle 0\) Definition of Addition in Peano Structure

which is a contradiction of $(P4)$: $0$ is not in the image of $s$.


Case 2: $k < b$ and $s \left({k}\right) = b$

$(1')$ is true

This follows from the assumption.


$(2')$ is false

Aiming for a contradiction, suppose $(2')$ is true.

Let $c$ be a non-zero natural number such that:

$s \left({k}\right) = b + c$


Then:

\(\displaystyle s \left({k}\right)\) \(=\) \(\displaystyle b + c\) Assumption
\(\displaystyle b\) \(=\) \(\displaystyle b + c\) Assumption
\(\displaystyle 0\) \(=\) \(\displaystyle c\) Natural Number Addition is Cancellable

which is a contradiction of our assumption that $c$ is non-zero.


$(3')$ is false

Aiming for a contradiction, suppose $(3')$ is true.

Let $c$ be a non-zero natural number such that:

$s \left({k}\right) + c = b$


Then:

\(\displaystyle s \left({k}\right) + c\) \(=\) \(\displaystyle b\) Assumption
\(\displaystyle b + c\) \(=\) \(\displaystyle b\) Assumption
\(\displaystyle c\) \(=\) \(\displaystyle 0\) Natural Number Addition is Cancellable

which is a contradiction of our assumption that $c$ is non-zero.


Case 3: $k < b$ and $s \left({k}\right) \ne b$

From the definition of $<$, we have the existence of a non-zero natural number $m$ such that:

$k + m = b$

By Non-Successor Element of Peano Structure is Unique, there exists a natural number $p$ such that:

$s \left({p}\right) = m$

Substituting:

$k + s \left({p}\right) = b$

Applying the definition of addition in Peano structure, we get:

$s \left({k + p}\right) = b$


$(1')$ is false

It is assumed in this case that:

$s \left({k}\right) \ne b$

Therefore, $(1')$ is false.


$(2')$ is false

Aiming for a contradiction, suppose $(2')$ is true.

Let $c$ be a non-zero natural number such that:

$s \left({k}\right) = b + c$

By Non-Successor Element of Peano Structure is Unique, there exists a natural number $d$ such that:

$s \left({d}\right) = c$


Then:

\(\displaystyle s \left({k}\right)\) \(=\) \(\displaystyle b + c\) Assumption
\(\displaystyle \) \(=\) \(\displaystyle \left({k + s \left({p}\right)}\right) + s \left({d}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle k + \left({s\left({p}\right) + s \left({d}\right)}\right)\) Natural Number Addition is Commutative
\(\displaystyle \) \(=\) \(\displaystyle k + s\left({p + s \left({d}\right)}\right)\) Definition of Addition in Peano Structure
\(\displaystyle \) \(=\) \(\displaystyle s\left({k + \left({p + s \left({d}\right)}\right)}\right)\) Definition of Addition in Peano Structure
\(\displaystyle \) \(=\) \(\displaystyle s\left({k + \left({s \left({p + d}\right)}\right)}\right)\) Definition of Addition in Peano Structure
\(\displaystyle k\) \(=\) \(\displaystyle k + \left({s \left({p + d}\right)}\right)\) $(P3)$$:$ $s$ is injective
\(\displaystyle 0\) \(=\) \(\displaystyle s \left({p + d}\right)\) Natural Number Addition is Cancellable

which is a contradiction of $(P4)$: $0$ is not in the image of $s$.


$(3')$ is true

We have:

\(\displaystyle s \left({k + p}\right)\) \(=\) \(\displaystyle b\)
\(\displaystyle s \left({p + k}\right)\) \(=\) \(\displaystyle b\) Natural Number Addition is Commutative
\(\displaystyle p + s \left({k}\right)\) \(=\) \(\displaystyle b\) Definition of Addition in Peano Structure
\(\displaystyle s \left({k}\right) + p\) \(=\) \(\displaystyle b\) Natural Number Addition is Commutative

The result follows.


Case 4: $k > b$

From the definition of $>$, we have the existence of a non-zero natural number $m$ such that:

$k = b + m$

By Non-Successor Element of Peano Structure is Unique, there exists a natural number $p$ such that:

$s \left({p}\right) = m$

Substituting:

$k = b + s \left({p}\right)$

By applying $s$ to both sides, we obtain:

$s \left({k}\right) = s \left({b + s \left({p}\right)}\right)$

Applying the definition of addition in Peano structure, we get:

$s \left({k}\right) = b + s \left({s \left({p}\right)}\right)$


$(1')$ is false

Aiming for a contradiction, suppose $(1')$ is true.

Then:

\(\displaystyle s \left({k}\right)\) \(=\) \(\displaystyle b\) Assumption
\(\displaystyle b + s \left({s \left({p}\right)}\right)\) \(=\) \(\displaystyle b\)
\(\displaystyle s \left({s \left({p}\right)}\right)\) \(=\) \(\displaystyle 0\) Natural Number Addition is Cancellable

which is a contradiction of $(P4)$: $0$ is not in the image of $s$.


$(2')$ is true

We have:

$s \left({k}\right) = b + s \left({s \left({p}\right)}\right)$

The result follows.


$(3')$ is false

Aiming for a contradiction, suppose $(3')$ is true.

Let $c$ be a non-zero natural number such that:

$s \left({k}\right) + c = b$


Then:

\(\displaystyle s \left({k}\right) + c\) \(=\) \(\displaystyle b\) Assumption
\(\displaystyle \left({b + s \left({s \left({p}\right)}\right)}\right) + c\) \(=\) \(\displaystyle b\)
\(\displaystyle b + \left({c + s \left({s \left({p}\right)}\right)}\right)\) \(=\) \(\displaystyle b\) Natural Number Addition is Associative and Natural Number Addition is Commutative
\(\displaystyle \left({c + s \left({s \left({p}\right)}\right)}\right)\) \(=\) \(\displaystyle 0\) Natural Number Addition is Cancellable
\(\displaystyle s \left({c + s \left({p}\right)}\right)\) \(=\) \(\displaystyle 0\) Definition of Addition in Peano Structure

which is a contradiction of $(P4)$: $0$ is not in the image of $s$.

$\blacksquare$


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