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