Axioms, interpretations and relative consistency

Try

Take three points named A, B and C and declare each two-point subset to be a line. Every two distinct points determine exactly one line, just as in an elementary geometry classroom. Yet there is no fourth point and no line contains three points. This small structure already separates an incidence statement from the much richer geometry suggested by a drawing.

A geometric axiom system specifies primitive objects and relations, then statements those relations must satisfy. In this lesson “point,” “line” and “lies on” are primitive. A model assigns actual sets and relations to these words. A theorem must hold in every model of the assumptions used.

Predict

Predict whether “two distinct lines intersect at most once” needs distances, angles or parallels. It needs none of them. Unique joining lines suffice. Predict whether “two distinct lines intersect” follows as well. It does not: two parallel real coordinate lines supply a counterexample.

The difference between “at most one” and “exactly one” is a missing existence claim. Keep the quantifiers visible. A sketch often inserts an intersection automatically even when no axiom supplies one.

Experiment

Audit three structures by listing their objects rather than drawing approximately straight strokes.

Structure	Points	Lines	Unique join?
Three-point model	A, B, C	AB, AC, BC	Yes
Real affine plane	Ordered pairs of reals	Affine lines	Yes
One-line structure	A, B, C	ABC only	Yes, but no noncollinear triple

In the last row every pair belongs to the sole line. Thus unique joining lines alone do not ensure a genuinely two-dimensional geometry. Add an axiom that three noncollinear points exist and check which rows survive. This is a finite inspection of the named structures, not an enumeration of all geometries.

Understand

Here is a complete first incidence argument. Assume distinct points P and Q belong to distinct lines l and m. By the unique-join axiom there is exactly one line through P and Q. Both l and m meet that description, so l=m, contradicting their distinctness. Therefore two distinct lines have at most one common point.

Notice the proof’s architecture: hypotheses identify two allegedly shared points; uniqueness identifies the lines; contradiction eliminates the alleged configuration. No metric or order enters. The conclusion consequently holds in affine and projective incidence models alike.

A countermodel works in the opposite direction. To refute “every line has at least three points” as a consequence of unique joins, verify the three-point model satisfies the assumption and exhibit its two-point line AB. Checking only that AB is short is insufficient unless the background axiom is also verified.

Apply

Repair the argument “The ordinary plane satisfies the parallel postulate, so incidence proves it.” One example establishes satisfiability, not consequence in every model. A derivation must explain why the statement follows without silently importing a property of one preferred interpretation.

For optional logical depth, distinguish consistency from relative consistency. Constructing a geometry inside a real-number theory transfers consistency from that background: a contradiction in the interpreted geometry would yield a contradiction in the background. It does not establish the background’s absolute consistency. Likewise two models with opposite truth values for P establish independence only after all shared axioms have been checked.

Continue

Keep a proof ledger with four fields: primitives, axioms invoked, conclusion, and models tested. Later lessons add order, congruence and continuity in separate blocks, making hidden dependencies easier to find.

This undergraduate emphasis follows the axioms-and-models direction of UCLA’s mathematics course outlines. The examples and exercises here are original. Continue to finite incidence planes to see algebra supply complete small models.