A confusion about the second connection axiom of Euclidean Geometry

As far as I can tell, this is best described as an error in the text, possibly one introduced by the translator. These statements appear to be going to some contortions to state everything in terms of the notation "$AB=a$", which is not used in the original German (which you can find in pdf form here). This is all rather awkwardly done, and the explanation is so sketchy (what does "$AB=a$" actually mean and how is it related to "$A$ lies on $a$"??) that I would consider it "not even wrong". In particular, you are correct that the two halves of Axiom I,2 do not have the same meaning; the correct version of this axiom is the first half, not the second half.

So, I would advise that you completely ignore the second part of Axiom I,2. A more modern presentation of these two axioms (with their intended meanings) together with a definition of the notation $AB$ would be:

I,1. For any two points $A$ and $B$, there exists a line that contains both of them.

I,2. A line is uniquely determined by any two distinct points that lie on it. That is, if $A\neq B$ and $a$ and $b$ both contain $A$ and $B$, then $a=b$.

If $A\neq B$, we write $AB$ for the unique line which contains both $A$ and $B$. (Such a line exists by I,1 and is unique by I,2.)

Here is some elaboration on how incoherent and awful the presentation of the axioms you quoted is. Axiom I,1 is rather ambiguous: what does "completely determine" mean? By the usual English meaning of this phrase, it would include a statement of uniqueness, so it would seem to say for any two points $A$ and $B$, there is a unique line containing them both. But if this is the intended meaning, then Axiom I,2 is completely redundant. So, it seems that despite its phrasing, Axiom I,1 is not intended to include any assertion of uniqueness (and indeed other presentations of Hilbert's axioms convert Axiom I,1 into a statement just about existence as I have above).

As for the meaning of $AB=a$, I can think of three reasonable interpretations. Interpretation 1 is that $AB=a$ means "$A$ and $B$ both lie on $a$". Interpretation 2 is that $AB=a$ means "$a$ is the unique line that contains both $A$ and $B$". Interpretation 3 is that $AB=a$ is a primitive notion, defining a function $f(\{A,B\})=AB$ from unordered pairs of distinct points to lines (and this function is what is meant by "completely determine").

All three of these interpretations are problematic. Interpretation 1 turns the second half of Axiom I,2 into an immediate consequence of the definition: if $AB=a$ and $AC=a$, then in particular $B$ and $C$ both lie on $a$, so $BC=a$. Interpretation 2 is rather odd if Axiom I,1 is not intended to include a statement of uniqueness, given that the notation $AB=a$ is presented part of Axiom I,1. Moreover, if Axiom I,1 does not include a uniqueness statement, interpretation 2 makes the second part of Axiom I,2 worthless, because it cannot be used unless you already know that $AB=a$ and $AC=a$ and no axiom guarantees that this will ever happen. Interpretation 3 leaves it completely mysterious what this primitive notion $AB=a$ has to do with the other primitive notion "$a$ contains $A$", and in any case the two halves of Axiom I,2 still do not have the same meaning.

Of course, if you ignore the second half of Axiom I,2 and take the first half as the intended uniqueness statement as I wrote it above, then interpretations 1 and 2 become equivalent and unproblematic.

If $AB=a$ and $AC=a$, with $B\ne C$, then you know from Axiom 1 that points $BC$ lie on some line: Axiom 2 states that such line is still $a$.

So I think the words added after "that is" are a way to better explain the meaning of this axiom, while your $AB=AC=BC=a$ would be a mere repetition of the first part of the axiom.


To avoid confusion, one should keep in mind that Hilbert does not define lines and planes as sets of points. He writes:

Let us consider three distinct systems of things. The things composing the first system, we will call points [...]; those of the second, we will call straight lines [...]; and those of the third system, we will call planes [...]. We think of these points, straight lines, and planes as having certain mutual relations, which we indicate by means of such words as “are situated,” “between,” “parallel,” “congruent,” “continuous,” etc. The complete and exact description of these relations follows as a consequence of the axioms of geometry.

The first two axioms are then devoted to describe the relations between points and lines and they are both needed, in my opinion.