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Euclid had his 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

1 Answer 1

The axioms of Euclid are insufficient, by a fair distance, for proving the theorems in Elements. The first mistake is in the first proposition, which is essentially about drawing an equilateral triangle.

The procedure involves drawing certain circles. Then the equilateral triangle has vertices at certain places where the circles meet. But Euclid does not show that the circles indeed do meet.

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.

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