# What does the completeness of a system mean for the provability of certain statments?

Through the 16th to 19th centuries mathematicians tried to prove the Euclids parallel postulate from Euclid's other four postulates. In the beginning of the 19th century the mathematics community found that the fifth postulate is logically independent of the other four, and thus could be replaced another postulate, giving rise to non-euclidean geometry.

In the 20th century Tarski provided an axiomatic formulation of the euclidean geometry that is complete.

This leads me to a few questions. If Euclid's fifth postulate was unprovable from the other four, wouldn't that make (euclidean geometry - parallel postulate) incomplete? After all, there is an undecidable proposition. That would seemingly make the entirety of Euclidean geometry incomplete.

Taking this further, couldn't you remove an axiom from any axiomatic system, and assuming that axiom is not redundant, then you have a truth that is no longer provable? (otherwise you wouldn't need it as an axiom in the first place.)

I think I am mistaken about the nature of Gödel's theorems, completeness, decidability, or all of them. I was hoping someone could set me straight.

• The traditional Euclidean axiomatization, with the Fifth Postulate, was not an axiomatization in the modern sense. And it was quite incomplete. The notion of betweenness was used informally, and many (most?) of the theorems were not really proved. For example, at the very beginning an equilateral triangle is constructed, the usual circles construction. But nowhere is it proved that the circles meet. Hilbert (second order) and Tarski (first order) gave the first correct axiomatizations. – André Nicolas Jul 11 '12 at 4:37