This is math/philosophical question. Are all problems solvable? By solution, I also mean that if a problem has no solution, then that is still a solution. What I mean is that for every problem, is there always a solution? I was inspired to think about this after reading about the P vs NP problem. An example would be $x>x-1$ The fact that there is no solution for $x$ is, in fact, a solution for the problem as a whole.
2026-03-27 00:02:00.1774569720
On
Are all questions solvable?
1.3k Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
2
There are 2 best solutions below
2
On
No. The Halting Problem is the quintessential example.
I don't understand Godel's Incompleteness Theorem, but it's also worth a look if you're interested in these sorts of questions.
It is known that there are polynomial equations in several variables with integer coefficients where it is impossible to determine in any systematic way whether or not there is an integer solution where one of the variables has a particular value. (This comes out of Hilbert's 10th problem).
But knowing that there is no solution is a solution too, you say? Well, how about the fact that it is also impossible to know for sure whether or not the equation you're looking at is one of the "impossible" ones?
We can't declare everything to be solvable just by moving goalposts. Depending on the precise meaning you choose for "problem" and "solution" (and even what you take it to mean that "there is" a solution), you'll be able to stave off the admission of unsolvability for some small number of steps by creative goalpost placement -- but sooner or later the buck has to stop.