How to prove that the 20 axioms of geometry, in Hilbert's axiomatic system, are independent of each other?
In other words, based on a logic-theory statement, which are the worlds in which 19 of the axioms are true and the other one left is not? For example, how to prove that the 7th incidence axiom("If two planes α, β have a point A in common, then they have at least a second point B in common.") is independent of the other 19 axioms?
To better understand my question, here's an analogy:
In group theory, we prove that associative and commutative laws are independent of each other by taking a group that is associative but not commutative(Matrices) and a group that is commutative but not associative($x*y = x+y – xy$, for each $x$ and $y$ in $\mathbf{R}$ (the reals)).
Link about Hilbert's axioms:
That is exactly how we proceed, and so to prove that all 20 axioms are needed, we would need 20 different all-but-one-axiom models.
Some of them are easy to construct. For example, if we want a model of all the axioms except I.7, we can take ordinary Euclidean space and add a plane that only includes a single point $P$.
Others are harder. Notably, there's the non-Euclidean geometries in which all axioms except IV.1 hold. Non-Archimedean geometries in which V.1 fails to hold also take some thinking to find. To do these, we replace $\mathbb R$ by a larger, non-Archimedean field, where we add a variable $t$ and all functions obtained from $t$ by addition, multiplication, division, and the map $x \mapsto \sqrt{1+x^2}$.
Hilbert's Foundations of Geometry gave some of these models, but I'm not sure if it includes all 20 of them.