consistency

What is the worst thing that can happen to a set of axioms? Not that it is uninteresting, but that it secretly contradicts itself. If from your axioms you can prove both a statement and its negation — both 'lines AB and CD meet' and 'lines AB and CD do not meet' — then by the rules of logic you can go on to prove absolutely everything, true or false, and the whole system becomes worthless. A consistent axiom system is one that escapes this fate: it is impossible to derive a contradiction from it. Consistency is the minimum a system must have to mean anything at all.

How could you ever be sure no contradiction lurks, when there are infinitely many proofs you cannot check one by one? The standard tool is a model. You exhibit some concrete mathematical structure — points, lines, and relations built from objects you already trust, such as real numbers — and verify that every axiom comes out true in that structure. If the axioms were contradictory, the contradiction would have to show up as a false statement inside the model; but a real, existing structure cannot contain a genuine contradiction. So a model demonstrates consistency. For Euclidean geometry the model is the coordinate plane of pairs of real numbers, with 'line' meaning a linear equation — every Hilbert axiom can be checked there.

There is an honest catch, made precise by Godel. A model only proves relative consistency: Euclidean geometry is consistent provided arithmetic of the real numbers is consistent, because the model is built from real numbers. You have not proved consistency from nothing — you have reduced trust in geometry to trust in numbers. Godel's second incompleteness theorem says this is unavoidable: a sufficiently rich consistent system can never prove its own consistency from within. So consistency is always argued by leaning on a system you already accept, never conjured out of pure air.