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it's almost an antithetical non-sequitur at that, because Euclidean Geometry is consistent.

hinkley 8y ago
“and non Euclidean geometry”. You are reading that as two statements instead of a compound statement. Euclidean geometry is self consistent but not consistent with general relativity. The triangle inequality doesn’t hold when there is a large mass near the hypotenuse. In a less extreme example, it also doesn’t hold for real world problems of the Traveling Salesman sort (one way roads and steep hills). It’s really good for flat surfaces. But we don’t live on a flat surface.
- TeMPOraL 8y ago
  > Euclidean geometry is self consistent but not consistent with general relativity. The triangle inequality doesn’t hold when there is a large mass near the hypotenuse. What has one to do with the other? Am I missing some important context here, because the statement "euclidean geometry is not consistent with general relativity" doesn't make any sense to me. Euclidean geometry is mathematical abstraction built out of axioms that were chosen to sorta match our intuitive understanding of space. Relativity is a physics theory that uses some different mathematical geometry built out of different axioms, chosen so that resulting geometry would better fit observational data. And Gödelian consistency has nothing to do with physics at all, it considers mathematical constructs.
  - whatshisface 8y ago
    I think that consistency can matter to physicsts, there are some theories that can be written as axiomatic postulates that may be shown to contradict, not contradict, or be equivalent to ZFC as far as that question is concerned.
- andrepd 8y ago
  It's mathematics. Not physics. You're mixing things on a very bad level. Mathematics exists in the mind, so to speak. There is no requirement of consistency with the physical world, even though some mathematics may be a useful language for modelling our physical theories of reality. Euclidean geometry is self-consistent, and so is non-euclidean geometry. The point I think they were trying to make is that the parallel postulate is independent of the other 4, something that quite flustered mathematicians for millennia, until it was discovered that changing the parallel postulate led to different geometries, also very rich and interesting. So the parallel postulate is an example of an unprovable proposition in geometry with the other 4 axioms.
  - 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 ago
      Godel's theorem doesn't apply to Euclidean geometry. Tarski proved it is complete, consistent, and decidable.
      - LolWolf 8y ago
        Yes, 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).
- LolWolf 8y ago
  I think there are some crossed wires here. "Consistency" in this sense does not talk about how two systems are related to each other[0]—it is a question of whether you can prove that a system of axioms, call it S, is not absurd (i.e., cannot produce the statement, say, 0=1). Godel's incompleteness answers the question, "can we prove S is consistent, assuming only the axioms given in S?" with a negative, assuming S is a complicated-enough set of axioms.[1] ---- [0] The usual disclaimer holds, we can talk about proving the consistency of some axiomatic system A, if some other axiomatic system B is consistent. E.g., we can show that the integers with the usual ring structure is consistent if ZFC is consistent, because we can construct integer arithmetic from ZFC. This, though, is a point I avoid for clarity since this is distinct from what I believe is being mentioned. [1] I.e., if it can embed the rules of natural-number arithmetic in some way.

“and non Euclidean geometry”. You are reading that as two statements instead of a compound statement. Euclidean geometry is self consistent but not consistent with general relativity. The triangle inequality doesn’t hold when there is a large mass near the hypotenuse. In a less extreme example, it also doesn’t hold for real world problems of the Traveling Salesman sort (one way roads and steep hills). It’s really good for flat surfaces. But we don’t live on a flat surface.

TeMPOraL 8y ago
> Euclidean geometry is self consistent but not consistent with general relativity. The triangle inequality doesn’t hold when there is a large mass near the hypotenuse. What has one to do with the other? Am I missing some important context here, because the statement "euclidean geometry is not consistent with general relativity" doesn't make any sense to me. Euclidean geometry is mathematical abstraction built out of axioms that were chosen to sorta match our intuitive understanding of space. Relativity is a physics theory that uses some different mathematical geometry built out of different axioms, chosen so that resulting geometry would better fit observational data. And Gödelian consistency has nothing to do with physics at all, it considers mathematical constructs.
- whatshisface 8y ago
  I think that consistency can matter to physicsts, there are some theories that can be written as axiomatic postulates that may be shown to contradict, not contradict, or be equivalent to ZFC as far as that question is concerned.
andrepd 8y ago
It's mathematics. Not physics. You're mixing things on a very bad level. Mathematics exists in the mind, so to speak. There is no requirement of consistency with the physical world, even though some mathematics may be a useful language for modelling our physical theories of reality. Euclidean geometry is self-consistent, and so is non-euclidean geometry. The point I think they were trying to make is that the parallel postulate is independent of the other 4, something that quite flustered mathematicians for millennia, until it was discovered that changing the parallel postulate led to different geometries, also very rich and interesting. So the parallel postulate is an example of an unprovable proposition in geometry with the other 4 axioms.
- 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 ago
    Godel's theorem doesn't apply to Euclidean geometry. Tarski proved it is complete, consistent, and decidable.
    - LolWolf 8y ago
      Yes, 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).
LolWolf 8y ago
I think there are some crossed wires here. "Consistency" in this sense does not talk about how two systems are related to each other[0]—it is a question of whether you can prove that a system of axioms, call it S, is not absurd (i.e., cannot produce the statement, say, 0=1). Godel's incompleteness answers the question, "can we prove S is consistent, assuming only the axioms given in S?" with a negative, assuming S is a complicated-enough set of axioms.[1] ---- [0] The usual disclaimer holds, we can talk about proving the consistency of some axiomatic system A, if some other axiomatic system B is consistent. E.g., we can show that the integers with the usual ring structure is consistent if ZFC is consistent, because we can construct integer arithmetic from ZFC. This, though, is a point I avoid for clarity since this is distinct from what I believe is being mentioned. [1] I.e., if it can embed the rules of natural-number arithmetic in some way.