Postulates that Euclid neglected to mention?

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SUMMARY

The discussion centers on the postulates of Euclidean geometry as presented in Euclid's Elements, specifically questioning whether mathematicians universally accept the five original postulates. The consensus indicates that while these five postulates are foundational, David Hilbert's work in "Foundations of Geometry" addresses limitations and proposes additional axioms for a more comprehensive framework. This suggests that the mathematical community recognizes the need for further exploration beyond Euclid's original propositions.

PREREQUISITES
  • Understanding of Euclidean geometry principles
  • Familiarity with Euclid's Elements
  • Knowledge of Hilbert's axiomatic system
  • Basic comprehension of mathematical logic and axioms
NEXT STEPS
  • Study Hilbert's "Foundations of Geometry" for insights on additional axioms
  • Explore the implications of non-Euclidean geometries
  • Research the historical context of Euclid's postulates and their acceptance
  • Investigate modern mathematical logic and its relation to geometry
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Mathematicians, educators, students of geometry, and anyone interested in the foundational aspects of mathematical theory.

LeonhardEuler
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Out of curiosity, I was recently reading through Euclid's Elements online (http://aleph0.clarku.edu/~djoyce/java/elements/elements.html" )

My question is, is this the general opinion of all mathematicians? Also, has anyone tried to fix this problem, and how many more postulates would they need? I'm really surprised because I thought for a long time that only those 5 postulates were needed to axiomize Euclidean geometry.
 
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It's generally accepted. Hilbert pretty much fixed it. See Foundations of Geometry, D. Hilbert (trans. L. Unger), Open Court Publ.
 
Thanks a lot, I'll do that!
 

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