Can Hilbert's Axioms for Geometry be Derived from Zermelo-Fraenkel Set Theory?

Hilbert's axioms for geometry introduce primitive undefined terms such as "line," whose truth is relative to a particular choice of what each term refers to. This fundamental aspect of Hilbert's axiomatic system raises interesting questions about the relationship between Hilbert's geometry and set theory, particularly the Zermelo-Fraenkel (ZF) set theory. In this article, we explore whether Hilbert's axioms for geometry can be derived from ZF set theory, or if they form a distinct definition for Euclidean geometry.

The Familiar Model of Euclidean Geometry

A familiar model of Euclidean geometry is the use of coordinate geometry, specifically R2 for the Euclidean plane and R3 for Euclidean three-dimensional space. This model demonstrates that these spaces are indeed models of the axioms of Hilbert's system. The proof that these models satisfy Hilbert's axioms can be shown within the framework of ZF set theory. However, it is worth noting that the use of ZF set theory is more powerful than necessary for this proof. For instance, the axiom of replacement could have been omitted, and Zermelo set theory would still suffice.

Hilbert's Axioms and Zermelo Set Theory

While ZF set theory can prove that Hilbert's axioms are satisfied in the context of coordinate geometry, it is important to recognize that Hilbert's axioms are not mere definitions that can be derived directly from set theory. Rather, they form a definition for Euclidean geometry. The axioms can be stated within a set-theoretic framework, but this is not their primary function.

Hilbert himself did not introduce sets until the completeness axiom, which literally refers to a "system," not a set. The Axiom of Completeness states: To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner that the system thus generalized shall form a new geometry obeying all of the five groups of axioms. In other words, the elements of geometry, whether defined or undefined, form a system that is not susceptible to extension without violating these axioms. This underscores the fact that the elements of geometry (such as lines, points, and planes) are not sets of points or other mathematical constructs, although they can be described in such terms.

The Role of Language in Geometry

When using Hilbert's axioms without referring to sets, a line is not considered to be a set of points. Similarly, angles and plane figures are not considered to be sets of points. The theorems derived from Hilbert's axioms are the same regardless of whether sets are used or not; the difference lies merely in the language and interpretation. Thus, while set theory can be used to describe geometric elements, it is not essential for the fundamental axioms of Euclidean geometry.

Conclusion

In summary, Hilbert's axioms for Euclidean geometry cannot be derived directly from Zermelo-Fraenkel set theory. While these axioms can be expressed within the language of set theory, they serve a more fundamental purpose as a definition for Euclidean geometry. The axioms establish a consistent and complete system, independent of set-theoretic formulations, which forms the basis for the theorems and properties of Euclidean geometry. Understanding this distinction is crucial for appreciating the foundational nature of Hilbert's axioms in geometry.