Suppose we adopt all of Hilbert's axioms except Side-Angle-Side (SAS):
IV-6. If, in the two triangles ABC and A′B′C′ the congruences AB ≅ A′B′, AC ≅ A′C′, ∠BAC ≅ ∠B′A′C′ hold, then the congruences ∠ABC ≅ ∠A′B′C′ and ∠ACB ≅ ∠A′C′B′ also hold.
Are the following axioms equivalent?
- 1. Constant curvature
- 2. Congruence is preserved by translations.
- 3. Congruence is preserved by rotations.
- 4. Congruence is preserved by reflections.
- 5. If the two short legs of two right triangles are congruent, then the two triangles are congruent.
- 6. If the two short legs of two isosceles right triangles are congruent, then the two triangles are congruent.
- 7. If the two short legs of two isosceles right triangles are congruent, so is the hypotenuse of each.
- 8. Side-side-side (SSS). This is already known to be equivalent to SAS as an axiom.
One reason this is interesting is that so many interesting axioms are equivalent, just as it is interesting how many ways the parallel postulate can be stated. The parallel postulate is, I think, that curvature is zero.
I dislike SAS as an axiom. It looks like a theorem. It is not simple. SSS, which is an equivalent axiom, is better. What I would really like is axiom 7 above, though, which is, I think, a fair restatement of Euclid's axiom that "All right angles are equal". If that can be shown, then we have rescued Euclid from his most obvious and serious flaw, reliance on his silly principle that equal shapes are equal (which he uses to prove SAS in I-4 and SSS in I-8), without having to add any new axioms. In fact, we have reduced his system by one, since we can drop the silly principle.
Note that Hilbert's other axioms give us things like Euclid's "no holes" and "lines can be drawn infinitely long" (maybe overlapping onto themselves, on a sphere for example). Probably this is the same as saying we assume there is a C-0 infinite manifold, or something like that. We need some kind of infinite metric space. For the equivalences above, I don't think we need all the Hilbert axioms fixing up Euclid's omissions, though maybe we need some. We need a good definition of "angle" for example. Maybe we need some Incidence axioms. I don't think we need Pasch's Axiom or any Intersection axioms (e.g. the axiom needed in Euclid I-1 to prove that overlapping circles intersect). Maybe we need Hilbert's Archimedean axiom to get very small rotations and suchlike.