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In 2D geometry, where the five axioms of Euclid are right, can I prove any geometric theorem? like pythagorean theorem, bisector theorem, law of parallelogram, etc.?

I think that it could, because the axioms are the basis of all this geometry and therefore, more complex theorems should be able to express themselves in terms of their "elementary particles" (analogy). But this is just an intuition.

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    $\begingroup$ I don’t think you are supposed to call something a theorem if nobody can prove it, so, tautologically, yes, theorems have proofs, if that’s what you mean. Whether or not the random assertions you have in mind are probable from the random axioms you have in mind is a completely different issue , one which we have insufficient data on to answer. $\endgroup$ – rschwieb Sep 24 '18 at 10:29
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    $\begingroup$ I think you are asking if every conceivable true statement about geometry can be proven from Euclid's axioms. The term for that is completeness - you want to know if Euclid's axioms are complete. $\endgroup$ – Jack M Sep 24 '18 at 12:08
  • $\begingroup$ To answer the question, I suspect the answer is no, by Godel's incompleteness theorem, because surely any reasonable formalization of Euclid's axioms would be able to talk about arithmetic (since Euclid's work involves notions of length and area). $\endgroup$ – Jack M Sep 24 '18 at 12:15
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No.

Euclid made a number of unstated assumptions, which the German mathematician David Hilbert recognized. Hilbert addressed them in his text Grundlagen der Geometrie (Foundations of Geometry) by adding new axioms. Among the omissions made by Euclid was the concept of what it means for an object to be inside a closed curve, a question answered by the French mathematician Camille Jordan, a result known as the Jordan curve theorem.

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  • $\begingroup$ I think when OP says "any" they mean "all", and not "at least one". Ie, are Euclid's axioms complete. In that case, the lack of perfect rigor in Euclid's axioms is really a technical detail, not relevant to the question. $\endgroup$ – Jack M Sep 24 '18 at 12:06
  • $\begingroup$ @JackM That is my understanding, too. As I stated, Euclid's axioms were not complete, which is why Hilbert had to add additional axioms. Therefore, it is not possible to prove any theorem based on Euclid's axioms. $\endgroup$ – N. F. Taussig Sep 24 '18 at 12:08
  • $\begingroup$ I don't think it's really meaningful to talk about whether Euclid's axioms are complete or consistent, precisely because they're so vague. They don't constitute a formal system in the sense of modern logic. As I understand it the main reason Hilbert wrote his axioms wasn't to make them logically complete, but to make them into a proper formal system. $\endgroup$ – Jack M Sep 24 '18 at 12:12

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