Mark Aston
January 2017.

Are all mathematical truths universal?

1 answer

No, and here is an explicit example due to Saharon Shelah and Alexander Soifer.

Recall the innocuous Axiom of Choice, the statement that the Cartesian product of non-empty subsets is non-empty. Nobody disputes this for a finite collection, but for an infinite collection, there are many who protest that "Just take one from each!" runs into the non-trivial objection of "How, exactly?" Indeed, it was shown by Kurt Godel and Paul Cohen in two papers two decades apart that there are consistent models obeying the Zermelo-Fraenkel axioms where the Axiom of Choice holds and where it does not. So we will never really know whether the Axiom of Choice is true or not, unless our entire mathematical edifice is rebuilt from scratch with a different set of foundational axioms.

But isn't all this distinction abstrusely academic? Surely for the real theorems that people care about, none of this should matter? Ah, but this is where the Shelah-Soifer result becomes interesting. They consider a very natural object, namely a graph with vertex set equal to the set of real numbers, with two vertices x and y adjacent if and only if x-y-√2 is a rational number. They show that the chromatic number of this graph is 2 if the Axiom of Choice holds, but uncountably infinite if it does not. For more details, see:

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