According to Greenberg's book of geometry it is sufficient to consider the axiom of Dedekind along with Hilbert's axioms (except of course for the Archimedian Principle and his Axiom of Completeness) in order for Euclidian Geometry to be categorical (in this case the only model is that of order pairs of numbers used in Analytic Geometry).

From this I can say that, because Dedekind's axiom of continuity is equivalent to Cantor's axiom (Nested Intervals Principle) and because any of them implies the Archimedian Principle but not the other way around then I guess Hilbert's Axiom of completeness is necessary in order to fix that "something" missing if the Archimedian property is assumed as an axiom instead of the others.

So, my problem is basically that Hilbert's axiom of completeness is pretty weird and I cannot imagine in what sense it can be used along with Archimedes' Axiom to obtain Dedekind's Axiom.

Now, this is what Hilbert says in his book:

This axiom gives us nothing directly concerning the existence of limiting points, or of the idea of convergence. Nevertheless, it enables us to demonstrate Bolzano’s theorem by virtue of which, for all sets of points situated upon a straight line between two definite points of the same line, there exists necessarily a point of condensation, that is to say, a limiting point. From a theoretical point of view, the value of this axiom is that it leads indirectly to the introduction of limiting points, and, hence, renders it possible to establish a one-to-one correspondence between the points of a segment and the system of real numbers.

These are the definitions I'm talking about:

Hilbert's Axiom of completeness: 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 form a system which is not susceptible of extension, if we regard the five groups of axioms as valid.

Dedekind's Axiom: Suppose that the set ${l}$ of all points on a line $l$ is the disjoint union $\Sigma_1\cup \Sigma_2$ of two nonempty subsets such that no point of either subset is between two points of the other.Then there exists a unique point $o$ on $l$ such that one of the subsets is equal to a ray of $l$ with vertex $o$ and the other subset is equal to the complement.

Cantor's Axiom: Let $\{\overline{A_nB_n}\}_{n\in N}$ be a sequence of segments on some straight line such that $\overline{A_{n+1}B_{n+1}}\subset \overline{A_{n}B_{n}}$ for all $n\in \mathbb{N}$. Then the intersection of such segments is not empty and there is a point $o$ that belongs to all of them.

Archimedes' Axiom: For any two segments $\overline{AB}$ and $\overline{OE}$ there is a positive integer $n\in \mathbb{N}$ such that $\overline{AB}<n\cdot\overline{OE}$.


Hilbert's completeness axiom is not a standard axiom because it is about the other axioms, it is rather a meta-axiom about the models of the other axioms. Giovanni argues that Hilbert's was aware of Dedekind completeness axiom and chose this completeness property V.2 rather than second order Dedekind cuts because V.2 does not imply Archimedes axiom: http://dx.doi.org/10.1387/theoria.4544

See also the review of the above paper by Pambuccian which links to interesting related papers: https://zbmath.org/?q=an:06179688


Roughly speaking, with the Archimedean axiom, one restricts to the line to subfields of the reals. Maximality says all the reals are there.


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