Relative proofs of consistency

In the last post, we defined and explained the importance of the property of consistency for an axiomatic system. The second half of Section II in Nagel and Newman's book describes ways of proving that an axiomatic system is consistent.

But first, note that the question of consistency of Euclidean geometry did not arise. Its axioms were supposed to describe the real world; and something that actually exists cannot be self-contradictory. In other words, existence (or truth) implies internal consistency.

The need for consistency proofs arose much later, with non-Euclidean geometries, which do not obviously model space as we experience it. Non-existence does not imply inconsistency. But any interesting abstract construct had better be internally consistent.

Second, checking the internal consistency of all of the theorems produced so far is typically not a valid proof of consistency, because (interesting) axiomatic systems generate an infinite number of theorems. The proof of consistency must guarantee that not a single theorem, including some that we have not yet produced and that we might never produce, contradicts any other theorem in the system.

One possible way to prove the consistency of an axiomatic system is model-based, where a model is a kind of interpretation.

Consistency of axiomatic systems

The "problem of consistency" is the topic of Section II of Nagel and Newman's book.  This section defines "consistency" and explains when and why it became an important property of axiomatic systems.

The oldest and most famous axiomatic system is that of Euclid, in which he systematized all of the knowledge of geometry (and more) available to him over two thousand years ago. Based on five axioms, Euclid was able to rigorously prove a very large number of known and new theorems (called "propositions" in his Elements). His axioms were supposed to be intuitively true. The first four axioms dealt with line segments, lines, circles and angles (see this Wikipedia entry) and have been viewed as self-evident. In contrast, the fifth axiom, which was equivalent to the following statement: "Through a point outside a given line, only one parallel to the line can be drawn" (page 9), was not intuitively true (apparently because the two lines involved extend to infinity in two directions, similarly to asymptotes).

Since this proposed axiom was not obviously true, many mathematicians tried to prove that it logically follows from the first four axioms. Only in the nineteenth century was it demonstrated that it is NOT possible to prove the parallel axiom from the first four axioms.

This proof that it is impossible to prove a given statement is a great precursor of Gödel's incompleteness theorems.