Diaconescu-Goodman-Myhill Theorem

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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