Tarski's Axioms - Discussion

Discussion

Starting from two primitive relations whose fields are a dense universe of points, Tarski built a geometry of line segments. According to Tarski and Givant (1999: 192-93), none of the above axioms is fundamentally new. The first four axioms establish some elementary properties of the two primitive relations. For instance, Reflexivity and Transitivity of Congruence establish that congruence is an equivalence relation over line segments. The Identity of Congruence and of Betweenness govern the trivial case when those relations are applied to nondistinct points. The theorem xy≡zz ↔ x=y ↔ Bxyx extends these Identity axioms.

A number of other properties of Betweenness are derivable as theorems including:

• Reflexivity: Bxxy ;
• Symmetry: Bxyz → Bzyx ;
• Transitivity: (Bxyw ∧ Byzw) → Bxyz ;
• Connectivity: (Bxyw ∧ Bxzw) → (Bxyz ∨ Bxzy).

The last two properties totally order the points making up a line segment.

Upper and Lower Dimension together require that any model of these axioms have a specific finite dimensionality. Suitable changes in these axioms yield axiom sets for Euclidean geometry for dimensions 0, 1, and greater than 2 (Tarski and Givant 1999: Axioms 8(1), 8(n), 9(0), 9(1), 9(n) ). Note that solid geometry requires no new axioms, unlike the case with Hilbert's axioms. Moreover, Lower Dimension for n dimensions is simply the negation of Upper Dimension for n - 1 dimensions.

When dimension > 1, Betweenness can be defined in terms of congruence (Tarski and Givant, 1999). First define the relation "≤" (where is interpreted "the length of line segment is less than or equal to the length of line segment "):

In the case of two dimensions, the intuition is as follows: For any line segment xy, consider the possible range of lengths of xv, where v is any point on the perpendicular bisector of xy. It is apparent that while there is no upper bound to the length of xv, there is a lower bound, which occurs when v is the midpoint of xy. So if xy is shorter than or equal to zu, then the range of possible lengths of xv will be a superset of the range of possible lengths of zw, where w is any point on the perpendicular bisector of zu.

Betweenness can than be defined as

The Axiom Schema of Continuity assures that the ordering of points on a line is complete (with respect to first-order definable properties). The Axioms of Pasch and Euclid are well known. Remarkably, Euclidean geometry requires just the following further axioms:

• Segment Construction. This axiom makes measurement and the Cartesian coordinate system possible—simply assign the value of 1 to some arbitrary line segment;

Let wff stand for a well-formed formula (or syntactically correct formula) of elementary geometry. Tarski and Givant (1999: 175) proved that elementary geometry is:

• Consistent: There is no wff such that it and its negation are both theorems;
• Complete: Every sentence or its negation is a theorem provable from the axioms;
• Decidable: There exists an algorithm that assigns a truth value to every sentence. This follows from Tarski's:
  • Decision procedure for the real closed field, which he found by quantifier elimination;
  • Axioms admitting of a (multi-dimensional) faithful interpretation as a real closed field.

Gupta (1965) proved the above axioms independent, Pasch and Reflexivity of Congruence excepted.

Negating the Axiom of Euclid yields hyperbolic geometry, while eliminating it outright yields absolute geometry. Full (as opposed to elementary) Euclidean geometry requires giving up a first order axiomatization: replace φ(x) and ψ(y) in the axiom schema of Continuity with x ∈ A and y ∈ B, where A and B are universally quantified variables ranging over sets of points.