# Diaconescu-Goodman-Myhill Theorem

## Theorem

The axiom of choice implies the law of excluded middle.

## Proof

Let $\mathbb B = \set {0, 1}$.

Let $p$ be a proposition.

Let the following two sets be defined:

- $A = \set {x \in \mathbb B: x = 0 \lor p}$
- $B = \set {x \in \mathbb B: x = 1 \lor p}$

where $\lor$ denotes the disjunction operator.

We have that:

- $0 \in A$

and:

- $1 \in B$

so both $A$ and $B$ are non-empty

Then the set:

- $X = \set {A, B}$

is a set of non-empty sets:

By the axiom of choice, there exists a choice function:

- $f: X \to \mathbb B$

since $\displaystyle \bigcup X = \mathbb B$.

There are four cases:

- $(1): \quad \map f A = \map f B = 0$

This means that $0 \in B$.

But for that to happen, $\paren {0 = 1} \vee p$ must be true.

So by Disjunctive Syllogism, $p$ is true.

- $(2): \quad \map f A = \map f B = 1$

This means that $1 \in A$.

Arguing similarly to case $(1)$, it follows that $p$ is true in this case also.

- $(3): \quad \map f A = 1 \ne \map f B = 0$

This means that $A \ne B$ (or otherwise $f$ would pick the same element).

But if $p$ is true, that means:

- $A = B = \mathbb B$

which is a contradiction.

Therefore in this case:

- $\neg p$

- $(4): \quad \map f A = 0 \ne \map f B = 1$

Using the same reasoning as in case $(3)$, it is seen that in this case:

- $\neg p$

So by Proof by Cases:

- $\paren {p \vee \neg p}$

That is the Law of Excluded Middle.

$\blacksquare$

## Source of Name

This entry was named for Radu Diaconescu, Noah D. Goodman and John R. Myhill.

## Also known as

The Diaconescu-Goodman-Myhill Theorem is also known as Diaconescu's Theorem and the Goodman-Myhill Theorem.

It is ever $\mathsf{Pr} \infty \mathsf{fWiki}$'s endeavour to attest a theorem to as many contributors as appropriate.

## Historical Note

The proof of the **Diaconescu-Goodman-Myhill Theorem** was first published in $1975$ by Radu Diaconescu.

It was later independently rediscovered by Noah D. Goodman and John R. Myhill and published in $1978$.

However, the first appearance of the result itself was in Errett Albert Bishop's $1967$ work *Foundations of Constructive Analysis*, where he set it as an exercise, without including a solution.

## Sources

- 1975: Radu Diaconescu:
*Axiom of Choice and Complementation*(*Proc. Am. Math. Soc.***Vol. 51**: pp. 176 – 178) www.jstor.org/stable/2039868

- 1978: N.D. Goodman and J. Myhill:
*Choice Implies Excluded Middle*(*Math. Log. Quart***Vol. 24**: p. 461)