# Are there simple models of Euclid's postulates that violate Pasch's theorem or Pasch's axiom?

While reading a paper1 about the history of modern logic, I learned that some opinions (about deductive/axiomatic mathematics) typically attributed to David Hilbert can be traced back to Moritz Pasch. After googling for Moritz Pasch, I was surprised to learn that he had found important implicit assumptions in Euclid missing from the axioms/postulates. I read on Wikipedia that both Pasch's theorem and Pasch's axiom cannot be derived from Euclid's postulates.

Are there simple models similar to elliptic and hyperbolic geometry for the parallel postulate that allow to illustrate this fact in a simple way?

1Jose Ferreiros: The Road to Modern Logic - An Interpretation, The Bulletin of Symbolic Logic, Vol. 7, No. 4 (Dec., 2001), pp. 441-484. Available here, JSTOR, DOI: 10.2307/2687794.

• It seems to me that whether Pasch's theorem is a theorem of plane geometry or not depends on what one considers plane geometry to be; the theorem depends critically on a notion of order which isn't even part of geometry as Euclid defines it. AFAIK the only less than/greater than comparison in Euclid's axioms is in the parallel postulate, where the angles 'less than right angles' define the side on which non-parallel lines meet, and it's not even clear that that can be used to define a linear notion of less than, greater than, or between. Commented Apr 5, 2013 at 22:12
• As such, violating Pasch's theorem seems as simple as defining a new 'between' relation of one's own and saying by fiat that "B between (A, C)" and "C between (B,D)" but not "B between (A,D)". To invalidate this sort of trivial counterexample then a more explicit definition of the linear-order relationship would be needed, and it seems plausible that any useful definition that correlates with an inherent notion of betweenness would then allow the derivation of Pasch's theorem. Commented Apr 5, 2013 at 22:15
• @StevenStadnicki OK, I see. I guess that's the reason why "Pasch's theorem" cannot be derived, but is less important than "Pasch's axiom". Commented Apr 5, 2013 at 22:28

There are no simple models. To violate Pasch's Axiom in a Hilbert-type setup, we need to use a discontinuous solution of the Cauchy functional equation $$f(x+y)=f(x)+f(y)$$. Such a discontinuous solution requires (part of) the Axiom of Choice.