# Equivalence of Well-Ordering Principle and Induction

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## Theorem

The Well-Ordering Principle, the Principle of Finite Induction and the Principle of Complete Finite Induction are logically equivalent.

That is:

- Principle of Finite Induction: Given a subset $S \subseteq \N$ of the natural numbers which has these properties:
- $0 \in S$
- $n \in S \implies n + 1 \in S$

- then $S = \N$.

- Principle of Complete Finite Induction: Given a subset $S \subseteq \N$ of the natural numbers which has these properties:
- $0 \in S$
- $\set {0, 1, \ldots, n} \subseteq S \implies n + 1 \in S$

- then $S = \N$.

- Well-Ordering Principle: Every non-empty subset of $\N$ has a minimal element.

## Proof

To save space, we will refer to:

- The Well-Ordering Principle as
**WOP** - The Principle of Finite Induction as
**PFI** - The Principle of Complete Finite Induction as
**PCI**.

### PFI implies PCI

Let us assume that the **PFI** is true.

Let $S \subseteq \N$ which satisfy:

- $(A): \quad 0 \in S$
- $(B): \quad \set {0, 1, \ldots, n} \subseteq S \implies n + 1 \in S$.

We want to show that $S = \N$, that is, the **PCI** is true.

Let $P \paren n$ be the propositional function:

- $P \paren n \iff \set {0, 1, \ldots, n} \subseteq S$

We define the set $S'$ as:

- $S' = \set {n \in \N: P \paren n \text { is true} }$

$P \paren 0$ is true by $(A)$, so $0 \in S'$.

Assume $P \paren k$ is true where $k \ge 0$.

So $k \in S'$, and by hypothesis:

- $\set {0, 1, \ldots, k} \subseteq S$

So by $(B)$:

- $k + 1 \in S$

Thus:

- $\set {0, 1, \ldots, k, k + 1} \subseteq S$.

That last statement means $P \paren {k + 1}$ is true.

This means $k + 1 \in S'$.

Thus we have satisfied the conditions:

- $0 \in S'$
- $n \in S' \implies n + 1 \in S'$

That is, $S' = \N$, and $P \paren n$ holds for all $n \in \N$.

Hence, by definition:

- $S = \N$

So **PFI** gives that $S = \N$.

$\Box$

### PCI implies WOP

Let us assume that the **PCI** is true.

Let $\O \subsetneqq S \subseteq \N$.

We need to show that $S$ has a minimal element, and so demonstrate that the **WOP** holds.

Aiming for a contradiction, suppose that:

- $(C): \quad S$ has no minimal element.

Let $\map P n$ be the propositional function:

- $n \notin S$

Suppose $0 \in S$.

We have that $0$ is a lower bound for $\N$.

Hence by Lower Bound for Subset, $0$ is also a lower bound for $S$.

$0 \notin S$, otherwise $0$ would be the minimal element of $S$.

This contradicts our supposition $(C)$, namely, that $S$ does not have a minimal element.

So $0 \notin S$ and so $\map P 0$ holds.

Suppose $\map P j$ for $0 \le j \le k$.

That is:

- $\forall j \in \closedint 0 k: j \notin S$

where $\closedint 0 k$ denotes the closed interval between $0$ and $k$.

Now if $k + 1 \in S$ it follows that $k + 1$ would then be the minimal element of $S$.

So then $k + 1 \notin S$ and so $\map P {k + 1}$.

Thus we have proved that:

- $(1): \quad \map P 0$ holds
- $(2): \quad \paren {\forall j \in \closedint 0 k: \map P j} \implies \map P {k + 1}$

So we see that **PCI** implies that $\map P n$ holds for all $n \in \N$.

But this means that $S = \O$, which is a contradiction of the fact that $S$ is non-empty.

So, by Proof by Contradiction, $S$ must have a minimal element.

That is, $\N$ satisfies the Well-Ordering Principle.

$\Box$

### WOP implies PFI

We assume the truth of **WOP**.

Let $S \subseteq \N$ which satisfy:

- $(D): \quad 0 \in S$
- $(E): \quad n \in S \implies n+1 \in S$.

We want to show that $S = \N$, that is, the **PFI** is true.

Aiming for a contradiction, suppose that:

- $S \ne \N$

Consider $S' = \N \setminus S$, where $\setminus$ denotes set difference.

From Set Difference is Subset, $S' \subseteq \N$.

So from **WOP**, $S'$ has a minimal element.

A lower bound of $\N$ is $0$.

By Lower Bound for Subset, $0$ is also a lower bound for $S'$.

By hypothesis, $0 \in S$.

From the definition of set difference, $0 \notin S'$.

So this minimal element of $S'$ has the form $k + 1$ where $k \in \N$.

We can consider the von Neumann construction of the natural numbers.

By definition of natural number addition, it is noted that $k + 1 \in \N$ is the immediate successor element of $k \in \N$.

Thus $k \in S$ but $k + 1 \notin S$.

From $(E)$, this contradicts the definition of $S$.

Thus if $S' \ne \O$, it has no minimal element.

This contradicts the Well-Ordering Principle, and so $S' = \O$.

So $S = N$.

Thus we have proved that **WOP** implies **PFI**.

$\Box$

### Final assembly

So, we have that:

- PFI implies PCI: The Principle of Mathematical Induction implies the Principle of Complete Induction
- PCI implies WOP: The Principle of Complete Induction implies the Well-Ordering Principle
- WOP implies PFI: The Well-Ordering Principle implies the Principle of Mathematical Induction.

This completes the result.

$\blacksquare$

## Sources

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- 1964: W.E. Deskins:
*Abstract Algebra*... (previous) ... (next): $\S 2.3$: Theorem $2.17$

- which demonstrates that PFI implies WOP

- 1964: W.E. Deskins:
*Abstract Algebra*... (previous) ... (next): $\S 2.3$: Theorem $2.18$

- which demonstrates that WOP implies PCI

- 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.17$: Finite Induction and Well-Ordering for Positive Integers: Theorem $17.3$

- which demonstrates that PFI and WOP are equivalent

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 10$: The well-ordering principle:

*Students familiar with the principle of induction can gain some sense of perspective in these matters by thinking out how the well-ordering principle can be proved from the principle of mathematical induction and vice versa.*

- 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 2.1$: The integers