Math CentralQuandaries & Queries


Question from Justin:

Hello there,

I was wondering is Euclid's Fifth Parallel Postulate of parallel lines never intersecting, undecidable or essentially undecidable?

Thank you so much for any help you can provide!




This is only meaningful in the context of the other axioms; and while Euclid's statement of the parallel axiom is clear, some of the others are not. (In particular, there is no axiom that allows us to deduce that two circles intersect.) So in what follows, we will assume that the other axioms have been fixed so that they do, collectively, what Euclid thought they did.

The parallel axiom does not state that parallel lines never intersect - that is the definition. Its original statement is rather complicated, but it is equivalent to the simpler "Playfair's Axiom" that states that there is a unique parallel to a line L through a point P not on L. (The existence of at least one parallel can be proved from the other axioms, so the real content of Playfair's axiom is that there cannot be more than one parallel.)

The parallel axiom is independent of the others, meaning that there are consistent geometries in which

  1. the other axioms and the parallel axiom hold [Euclidean geometry]

  2. the other axioms hold and the parallel axiom is false [hyperbolic geometry]

Amazingly, each of these models is unique up to isomorphism - there are no genuinely different geometries obeying axiom set (a) or axiom set (b)!

Moreover, within each geometry one can create a model of the other - so if Euclidean geometry is consistent, hyperbolic geometry MUST be consistent and vice versa!

Good hunting,


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