Julia Robinson helped define the limits of mathematical knowledge (2019)
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Re: Julia Robinson helped define the limits of mathematical knowledge (2019)
#2A nice article with a nice equation:
42 = -80538738812075974^3 + 80435758145817515^3 + 12602123297335631^3
Douglas Adams would rejoice. (checked it in bc)
Some Background from Wikipedia:
https://en.wikipedia.org/wiki/Diophantine_set#Matiyasevich's...
"Hilbert's tenth problem asks for a general algorithm deciding the solvability of Diophantine equations. The conjunction of Matiyasevich's result with the fact that most recursively enumerable languages are not decidable implies that a solution to Hilbert's tenth problem is impossible."
Re: Julia Robinson helped define the limits of mathematical knowledge (2019)
#3A nice article with a nice equation: 42 = -80538738812075974^3 + 80435758145817515^3 + 12602123297335631^3 Douglas Adams would rejoice. (checked it in bc) Some Background from Wikipedia: https://en.wikipedia.org/wiki/Diophantine_set#Matiyasevich's... "Hilbert's tenth problem asks for a general algorithm deciding the solvability of Diophantine equations. The conjunction of Matiyasevich's result with the fact that most…
The undecidability property proven here doesn't imply that there exists at least one Diophantine equation for which we'll never know if it's solvable or not, does it?
Re: Julia Robinson helped define the limits of mathematical knowledge (2019)
#4I was sorry to read that recently Martin Davis passed. Along with JR, another giant.
Re: Julia Robinson helped define the limits of mathematical knowledge (2019)
#5A nice article with a nice equation: 42 = -80538738812075974^3 + 80435758145817515^3 + 12602123297335631^3 Douglas Adams would rejoice. (checked it in bc) Some Background from Wikipedia: https://en.wikipedia.org/wiki/Diophantine_set#Matiyasevich's... "Hilbert's tenth problem asks for a general algorithm deciding the solvability of Diophantine equations. The conjunction of Matiyasevich's result with the fact that most…
The undecidability property proven here doesn't imply that there exists at least one Diophantine equation for which we'll never know if it's solvable or not, does it?
My understanding:
There's no algorithm to decide. But for any equation we can be lucky to find a solution or a proof that there's no solution.
But this doesn't prove that there is an equation for which we'll never know if it's solvable or not.
Re: Julia Robinson helped define the limits of mathematical knowledge (2019)
#6Earlier quoted context omitted.
The undecidability property proven here doesn't imply that there exists at least one Diophantine equation for which we'll never know if it's solvable or not, does it?
My understanding: There's no algorithm to decide. But for any equation we can be lucky to find a solution or a proof that there's no solution. But this doesn't prove that there is an equation for which we'll never know if it's solvable or not.
From my understanding, while that's is technically true, given a consistent axiomatic system (like ZFC[1], the foundation of mathematics we use) there exists a diophantine equation that can't be proven to have no solutions in that system (even though it has no solutions). This mathoverflow answer[2] gives the equation and a link to the paper that shows how to calculate the constants (the numbers are huge!).
What that means in practice is that although what you wrote is true, for some diophantine equations we'd have to come up with new axioms to be able to write a proof of the inexistence of its solutions. But then, how can we be sure that the the new axioms are consistent?
[1] I'm assuming ZFC is consistent; if it's not then it can prove anything, including the existence of solutions for any equations at all
Re: Julia Robinson helped define the limits of mathematical knowledge (2019)
#7A nice article with a nice equation: 42 = -80538738812075974^3 + 80435758145817515^3 + 12602123297335631^3 Douglas Adams would rejoice. (checked it in bc) Some Background from Wikipedia: https://en.wikipedia.org/wiki/Diophantine_set#Matiyasevich's... "Hilbert's tenth problem asks for a general algorithm deciding the solvability of Diophantine equations. The conjunction of Matiyasevich's result with the fact that most…
If anyone's interested in more context on that particular equation, here's an article I wrote for The Aperiodical at the time: https://aperiodical.com/2019/09/42-is-the-answer-to-the-ques...
Re: Julia Robinson helped define the limits of mathematical knowledge (2019)
#8A nice article with a nice equation: 42 = -80538738812075974^3 + 80435758145817515^3 + 12602123297335631^3 Douglas Adams would rejoice. (checked it in bc) Some Background from Wikipedia: https://en.wikipedia.org/wiki/Diophantine_set#Matiyasevich's... "Hilbert's tenth problem asks for a general algorithm deciding the solvability of Diophantine equations. The conjunction of Matiyasevich's result with the fact that most…
The undecidability property proven here doesn't imply that there exists at least one Diophantine equation for which we'll never know if it's solvable or not, does it?
In [0], Carl and Moroz give an explicit polynomial in 3639528+1 variables such that: a well-formed formula is a theorem in the first order predicate calculus if and only if the polynomial parametrized by the Diophantine coding of the formula (a single natural number) has a solution in N^{3639528}.
From this, they get an explicit Diophantine equation such that: the Godel-Bernays set theory is consistent if and only if that Diophantine equation has no solutions (and thus the same is true for ZFC, since NBG is a conservative extension of ZFC).
[0] https://link.springer.com/article/10.1007/s10958-014-1830-2
Re: Julia Robinson helped define the limits of mathematical knowledge (2019)
#9Interesting to learn she was the little sister of Constance Reid, who wrote a fantastic biography of Hilbert which is on my desk right now.
Re: Julia Robinson helped define the limits of mathematical knowledge (2019)
#10Earlier quoted context omitted.
My understanding: There's no algorithm to decide. But for any equation we can be lucky to find a solution or a proof that there's no solution. But this doesn't prove that there is an equation for which we'll never know if it's solvable or not.
From my understanding, while that's is technically true, given a consistent axiomatic system (like ZFC[1], the foundation of mathematics we use) there exists a diophantine equation that can't be proven to have no solutions in that system (even though it has no solutions). This mathoverflow answer[2] gives the equation and a link to the paper that shows how to calculate the constants (the numbers are huge!). What that…
Thanks.
I'm somewhat lost, but it seems to work Gödel like.
The statement is true (equation has no solution) but we can't prove it.