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What is (are) the most "elementary" question(s) one could ask in Euclidean geometry with all its postulates/axioms including the "5th" that is (are) known to be undecidable. The "5th" is undecidable by the rest but Gödel says even if we include it there will be something else. What would be the next simplest question one could ask in the axiom system with the 5th that cannot decided? I am looking for questions that are simple, intuitive, visualizable so I could show it to a 10 years old as an example.

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    $\begingroup$ You might want to take a look at en.wikipedia.org/wiki/Tarski%27s_axioms $\endgroup$ Commented May 5, 2021 at 14:52
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    $\begingroup$ "The "5th" is undecidable by the rest but Godel says even if we include it there will be something else." That's not true. Godel's theorem only applies to certain kinds of formal systems, and Euclidean geometry isn't such a system; indeed, Tarski's presentation of Euclidean geometry is complete. $\endgroup$ Commented May 5, 2021 at 14:53
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    $\begingroup$ Im not sure why a bounty was added. Sure more attention, but the comments resolve most imo and the rest is to be found easily. $\endgroup$ Commented Oct 16, 2024 at 21:39
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    $\begingroup$ @mick I am not sure either but OTOH according to en.wikipedia.org/wiki/Tarski%27s_axioms Tarski's system (of which I know nothing) "The system contains infinitely many axioms". That is not quite what I naively thought Euclidean geometry was about, to me it ended with Hilbert's system ... Maybe "user 2080", who put the bounty up, felt in a similar way. $\endgroup$ Commented Oct 16, 2024 at 22:28
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    $\begingroup$ This is kinda sorta a problem in plane geometry: Is there a way to color the plane with countably many colors so that there are no monochromatic triangles lined up with the x and y axes? Triangle here means just the three vertices, not the sides or interiors. The answer depends on the continuum hypothesis, so it's undecidable under ZFC. See risingentropy.com/… $\endgroup$ Commented Oct 20, 2024 at 12:34

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There are no such undecidable questions in Euclidean geometry. This is because, as others have pointed out, it's a decidable theory. Tarski’s system uses a single axiom schema for continuity that unfolds into infinitely many first-order instances. The Hilbert system uses just two axioms, but they need Second-order logic. The Wikipedia article points out that Euclidean geometry is not finitely axiomatizable in first-order logic. So that's the choice: a finite number of second-order axioms, or an infinite number of first-order axioms.

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  • $\begingroup$ Since my finitely and severely limited brain cannot comprehend an infinite number of axioms, see my comment above, the question is more related to something like Hilbert's system: what simple axiom may be missing from it? Thank you. $\endgroup$ Commented Oct 19, 2024 at 12:09
  • $\begingroup$ @hyportnex: So if we dig into it a bit more, Tarski’s system uses a single axiom schema for continuity that unfolds into infinitely many first-order instances. The Hilbert system uses just two axioms, but they need Second-order logic. The Wikipedia article points out that Euclidean geometry is not finitely axiomatizable in first-order logic. So that's the choice: a finite number of second-order axioms, or an infinite number of first-order axioms. $\endgroup$ Commented Oct 20, 2024 at 5:05
  • $\begingroup$ @hyportnex (To try a simple explanation). First-order logic allows quantization ($\forall$ and $\exists$) only for elements. Second-order logic allows quantization for sets of such elements, functions and relations between elements. When you have an axiom such as $\forall n \in \mathbb N, P(n)$, you can replace it by an infinity of non-quantized axioms: $P(0), P(1), P(2), ...$. Similarly, a second-order logic axiom can be replaced by an infinity of first-order logic axioms. $\endgroup$ Commented Oct 20, 2024 at 21:48

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