Let's remove one axiom from Euclidean geometry

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SUMMARY

This discussion explores the implications of removing axioms from Euclidean geometry, specifically focusing on the consequences of eliminating the postulate "To describe a circle with any center and distance." Historical attempts to remove the parallel postulate illustrate that such modifications lead to the development of new geometries. The conversation suggests that examining ordered geometry and absolute geometry may provide insights into the effects of altering fundamental axioms.

PREREQUISITES
  • Understanding of Euclidean geometry and its axioms
  • Familiarity with the parallel postulate and its historical context
  • Knowledge of ordered geometry concepts
  • Basic principles of absolute geometry
NEXT STEPS
  • Research the implications of removing the circle postulate in Euclidean geometry
  • Study the development and characteristics of non-Euclidean geometries
  • Explore the principles of ordered geometry in detail
  • Investigate the foundational aspects of absolute geometry
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Mathematicians, geometry enthusiasts, educators, and students interested in the foundational aspects of geometry and the implications of altering axiomatic systems.

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I'm wondering what could happen if we remove one axiom from Euclidean geometry. What are the conseqences? For example - how would space without postulate "To describe a cicle with any centre and distance" look like?
 
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The way you'd have to approach this is how specific theorems of Euclidean Geometry are affected when you remove a postulate.

http://mathworld.wolfram.com/EuclidsPostulates.html

As am example, there was a time when people tried to remove the parallel postulate and prove it from the other four but they all failed. Later it was determined that new geometries resulted from a relaxation of the parallel postulate.

https://en.wikipedia.org/wiki/Parallel_postulate

Off hand, I can't see the affects that removing the circle or allowing for multiple circles would have on the geometry. Perhaps looking at ordered geometry or absolute geometry will give you some insight:

https://en.wikipedia.org/wiki/Ordered_geometry

https://en.wikipedia.org/wiki/Absolute_geometry
 

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