AxiomIndependent of the axioms

Euclid's parallel postulate

Statement

If a straight line falling on two straight lines makes the interior angles on the same side sum to less than two right angles ( $\alpha + \beta < 180^\circ$ ), the two straight lines, if produced indefinitely, meet on that side. Equivalently (Playfair's axiom), in a plane, given a line $\ell$ and a point $P$ not on $\ell$ , there is at most one line through $P$ parallel to $\ell$ .

Why is it true?

Euclid's first four postulates describe local constructions — drawing segments, extending lines, drawing circles, and comparing right angles — that can be checked in a bounded region, whereas the fifth postulate makes a global claim about whether two lines with $\alpha + \beta < 180^\circ$ eventually meet arbitrarily far away. On a negatively curved saddle-like surface (the hyperbolic plane), geodesics spread apart so rapidly that infinitely many lines through $P$ never meet $\ell$ , showing that flatness at infinity is an independent choice of geometry rather than a logical consequence of local rules.

Proof sketch

The Cartesian plane $\mathbb{R}^2$ models all axioms of neutral geometry together with the parallel postulate, while the Beltrami–Klein and Poincaré disk models inside the unit disk $\mathbb{D} = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 < 1\}$ satisfy all axioms of neutral geometry together with the negation of the parallel postulate (infinitely many lines through $P$ disjoint from $\ell$ ). Because both models are built inside Euclidean geometry, neither the parallel postulate nor its negation can be deduced from the remaining axioms.

Stated by

stated -300Euclid of Alexandria

Topics that use this theorem

Non-Euclidean geometry

Related theorems

Gauss–Bonnet theorem Gauss's Theorema Egregium Thales's intercept theorem

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Euclid (translated by Thomas L. Heath) (1956). The Thirteen Books of Euclid's Elements, Vol. 1 (Book I, Postulate 5)
Marvin Jay Greenberg (2008). Euclidean and Non-Euclidean Geometries: Development and History
John Stillwell (1996). Sources of Hyperbolic Geometry · DOI:10.1090/hmath/010