Talk:Matiyasevich's theorem
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You wrote "Later work has shown that the question of solvability of a Diophantine equation is undecidable even if the number of variables is only 6". This is not accurate! Matijasevic showed that the number of variables over N={0,1,2,...} may be 9 (not 6), this appeared in J. P. Jones' paper in J. Symbolic Logic. And in 1992 Zhi-Wei Sun proved (in his Ph. D. thesis and some related papers) the undecidablity even if the number of variables over the integers is only 11.
- Could the anonymous person who wrote the words above edit the article itself? That's how Wikipedia works. Michael Hardy 02:20, 4 Dec 2003 (UTC)
Does anyone have a source for this statement? I'm curious what it's referring to.
- Matiyasevich's theorem has since been used to prove that many problems from calculus and differential equations are unsolvable.
Walt Pohl 18:55, 24 Oct 2004 (UTC)
"One can also derive the following stronger form of Gödel's incompleteness theorem from Matiyasevich's result: Corresponding to any given axiomatization of number theory, one can explicitly construct a Diophantine equation which has no solutions, but such that this fact cannot be proved within the given axiomatization."
Is there a logical mistake or could someone explain it to me: if one can explicitily construct an equation that has no solutions then he knows that there are no solutions so "the construction itself is a proof that this equation has no solutions". Shouldn't this statement be given as:
"Corresponding to any given axiomatization of number theory, there exists a Diophantine equation which has no solutions, but such that this fact cannot be proved within the given axiomatization." ?
MS 00:01, 03 Jun 2005