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Geometry/Is Euclid's Parallel Posulate Undecideable from the other Axioms of Euclidean Geometry

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Hey Azeem,

I appreciate your response, yeah they way I phrased it was wrong. I guess my question is can the Paralell Postulate be proved from the other axioms of Euclidean Geometry? Or is is independent of them and undecideable to prove it from them?

I hope this helps and thank you sooo much for your time and effort in answering this! I greatly appreciate it!

Sincerely,

Justin Thekkumthala

Answer
Hi Justin,

The parallel postulate cannot be proven using the other four axioms of Euclidean Geometry.  For centuries people tried, but it was eventually found that the parallel postulate is indeed necessary.

Also, be aware that "undecidable" does not mean the same thing as "cannot be proven from."  Your question
I guess my question is can the Paralell Postulate be proved from the other axioms of Euclidean Geometry?
a priori has nothing to do with decidability, only logical implication.

You could ask, "Given the four other axioms, decide whether the parallel postulate is true."  This would be a decision problem, which would be decidable because one could produce an algorithm that would always return "no." ...But I don't think that's what you were asking.
(This is a degenerate case because the input is always the same, so we wouldn't call this a decision problem.  It would just be a question, and the "algorithm" would be the logical/mathematical proof.)

Let me know if there's anything else,
Azeem

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Azeem Hussain

Expertise

I welcome your questions on algebra, 2D and 3D geometry, parabolic functions and conic sections, and any other mathematical queries you may have.

Experience

4 years as a drop-in and by-appointment tutor at Champlain College. Private tutor for dozens of clients over the past 8 years.

Publications
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry

Education/Credentials
Bachelor of Science, Major Mathematics and Major Economics, McGill University, 2014. Diploma of Collegiate Studies; Pure and Applied Science, Champlain College Saint-Lambert, 2010.

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