# What are the differences between Hilbert's axioms and Euclid's axioms?

Why would we need Hilbert's modern axiomatization of Euclidean geometry?

What are key differences between the two sets of axioms?

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Did you try to look up Hilbert's axioms? – GEdgar Mar 12 '13 at 2:02
There are really simple things that can't be proved via Euclidean postulates - they are hidden assumptions of Euclid. If I recall, the idea that a line is ordered is one of them - that if $a,b,c$ are three points on a line, then exactly one of the three is "between" the other two. – Thomas Andrews Mar 12 '13 at 2:12
read this: mathdl.maa.org/mathDL/22/… – Will Jagy Mar 12 '13 at 2:39
Hilbert saw the need for a completeness axiom. Say you have two points, $P$ and $Q$, and you draw a circle centered at $P$, and a circle centered at $Q$, both with radius $PQ$. How do you know that the two circles intersect? That there aren't "holes" in the plane where the intersections ought to be? Euclid's axioms don't do this for you; Hilbert's do. – Gerry Myerson Mar 12 '13 at 2:50

One major omission was noticed by Pasch, in the nineteeth century. Suppose that $ABC$ is a triangle, and let $\ell$ be a line that goes through a point $P$ on an edge of $\triangle ABC$. Then there is a point $Q$ (possibly equal to $P$,) on another edge of the triangle, such that $\ell$ passes through $Q$. The assumption that this is true is used tacitly in Elements. It is neither derived nor derivable from Euclid's axioms.
Around $1900$, Hilbert did a thoroughgoing axiomatization, with all details filled in. The result is vastly more complicated than the partial axiomatization by Euclid. One feature of the Hilbert axiomatization is that it is second-order. A benefit is that one can then prove that, for example, the Euclidean plane can be coordinatized using the real numbers.
Later, in the $1930$'s, Tarski produced an axiomatization that is first-order. Inevitably, we lose the result that the models are isomorphic to the coordinate plane $\mathbb{R}^2$. But there are bonuses, such as the later result (again by Tarski) that there is a decision procedure for elementary geometry.