Proving Parallel Postulate by showing that it is undecidable In the Numberphile video "Gödel's Incompleteness Theorem" (via YouTube), Professor Marcus Du Sautoy mentions that the Riemann hypothesis could be proved true by proving that it was undecidable, because it implies that no counterexample exists.

In a similar way, could you prove the parallel postulate is true (in an axiomatic system made up of the four other geometric axioms of Euclid) by showing that it is undecidable and consequently that no counterexample could exist?

Please try to explain this in as simple language as possible. I have very little experience in this area.
 A: Very briefly, no. The consequences of undecidability in both cases are (roughly) the same; there are models in which the statement is true and models in which the statement is false. The difference is in the character of those models (and what the statement being 'true' means). For the parallel postulate, 'day to day' Euclidean geometry provides a model where the postulate is true, but spherical geometry provides an equally good model of the other four of Euclid's postulates where the parallel postulate is false.
The situation with the Riemann hypothesis is potentially similar to this; arithmetic undecidability would imply that there are models of arithmetic in which it's true and models in which it's false. The difference is that for arithmetic we have a standard model, or in some sense a minimal model; the model consisting of 'just' the classic finite numbers. Now, consider an existential sentence along the lines of 'there exists a number $x$ such that...' — the presumption here is that the Riemann Hypothesis is equivalent to such a statement. Then if there are models in which this statement is true and models in which it's false, but those models agree on all the numbers in the standard model of arithmetic, then the $x$ that verifies the truth of the statement in a model where it's true must be a so-called 'non-standard' number, and this is what implies that in the standard minimal model the statement is false.
There's no such minimal model for Euclid's postulates, and that's one of the fundamental differences between the two situations. Another is the structure of the hypothesis — notice that being able to write the Riemann Hypothesis as a statement of existence is an essential piece of the argument that the thing it says exists can't exist in a minimal model (assuming undecidability). Isn't the parallel postulate also a statement about existence? Well, not exactly; what it says is that for all pairs of (line plus point not on that line), there exists another line such that etc. (And in fact, it asserts uniqueness of that other line, which is another wrinkle, but I'll skip that here). This is a more complicated statement than just the existence of a number, and to verify it we have to be able to look at all the things in the model.
