What is an example of a proof that explicitly relies on the law of excluded middle?

Question

I was talking with a friend about logic and I realized she might be an intuitionist. I was looking online for a proof that explicitly uses the law of excluded middle to see if she would have an issue with it, but we couldn't find anything she understood (she's not very good at math). The best example I could think of was the proof that an irrational number raised to an irrational power could be rational. She didn't understand it at all.

Can you think of an example of a proof that explicitly relies on the law of excluded middle that a lay person would probably understand? It doesn't have to be particularly mathematical in nature. In fact, the less math you assume she knows, the better.

Answer 1

Consider Euclid's theorem asserting that there are infinitely many prime numbers :

Consider any finite list of prime numbers $p_1, p_2,\ldots, p_n$. Let $P$ be the product of all the prime numbers in the list : $P = p_1p_2 \ldots p_n$ and let $q = P + 1$.

Then $q$ is either prime or not [...].

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See Euclid's original proof in Elements, Book IX, Prop.20 :

Let A, B, and C be the assigned prime numbers. I say that there are more prime numbers than A, B, and C.

Take the least number DE measured by A, B, and C [i.e. DE = least common multiple(A,B,C)].

Add the unit DF to DE.

Then EF is either prime or not.

Answer 2

See Colyvans article where he explicitly mentions it Colyvan, Mark, The philosophical significance of Cox's theorem, Int. J. Approx. Reasoning 37, No. 1, 71-85 (2004). ZBL1093.68113..

I think Professor Colyvan that it Cox's "derivation" of the "axioms of probability" makes use of the law of excluded middle.