6
votes
2answers
366 views

Gödel's (in)completeness theorems and the axiomatization of Euclidean geometry

In David Hilbert's 1899 Grundlagen der Geometrie, Hilbert gives a rigorous axiomatization of Euclidean geometry. As I understand it, some of Hilbert's axioms must be expressed in second order logic ...
2
votes
2answers
150 views

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 ...
8
votes
3answers
1k views

Why Euclidean geometry cannot be proved incomplete by Gödel's incompleteness theorems?

Wikipedia mentioned the limitation of Gödel's theorems. According to it, Euclidean geometry doesn't satisfy the hypotheses of Gödel's incompleteness theorems, that is, it cannot define natural ...