Looking for an original citation of Hilbert

Question

Do the axioms of complete ordered number fields come from the geometry Axioms of Hilbert? "The Foundations of Geometry (1899)" or did Hilbert publish some other axioms which evolved into the field axioms? I read this:

Hilbert had taken a totally different approach to defining the real numbers in 1900. He defined the real numbers to be a system with eighteen axioms. Sixteen of these axioms define what today we call an ordered field, while the other two were the Archimedean axiom and the completeness axiom.

The year seems not to line up 1899 vs 1900, and there were 21 geometry axioms as opposed to the 18 cited in the link. If there was a separate paper in 1900 defining the reals, does anyone have the original citation? I am not finding it, thanks.

Answer 1

See Foundations (1899) §13. COMPLEX NUMBER-SYSTEMS (page 23).

Hilbert enumerates 12 proerties of connection: $+, \cdot, 0, 1$, followed by 4 properties regarding order: $<$ and Archimedes axiom.

Up to now, 17 axioms.

Then he develops the so-called Algebra of segments, based on the plane geometry axioms (§24-on) and uses this "geometrical model" to show that:

theorems 1–6 of Section 13 are fulfilled. Moreover, [...] we have already shown that the laws 7–11 of operation, as given in Section 13, are all valid in this algebra of segments.

With the single exception of the commutative law of multiplication, therefore, all of the theorems of connection hold.

Then, in §28:

Upon the basis of the axioms of group II, we can easily show also that, in our algebra of segments, the laws 13–16 of operation given in Section 13 are fulfilled. Consequently, the totality of all the different segments forms a complex number system for which the laws 1–11, 13–16 of Section 13 hold; that is to say, all of the usual laws of operation except the commutative law of multiplication and the theorem of Archimedes.

Finally (§32), he proves the commutative law of multiplication.

The 18th axiom is the Axiom of Completeness (Vollständigkeit).