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Depending on what you mean, non-euclidean geometry may not be self-consistent (or it may! We don't know, unless we appeal to stronger axioms). In a large number
by LolWolf 8y ago
Depending on what you mean, non-euclidean geometry may not be self-consistent (or it may! We don't know, unless we appeal to stronger axioms). In a large number of cases, the set of axioms which produce the geometries in question are powerful enough to embed natural-number-like structures, which would imply that the consistency of the systems cannot be proven from within the systems, themselves.
- andrepd 8y agoGodel's theorem doesn't apply to Euclidean geometry. Tarski proved it is complete, consistent, and decidable.
- LolWolf 8y agoYes, that's fine. That Tarski's axioms for Euclidean geometry are consistent is independent of that claim—the above is a statement about the "usual" non-euclidean geometries (i.e., Riemannian).