How can a math problem be unsolvable?
In: Mathematics
One example of how a problem can be unsolvable is [undecidability](https://en.wikipedia.org/wiki/Undecidable_problem).
When a problem is undecidable, it means there is no method (program, algorithm, formula, logical statement, etc) that will always lead to a correct answer. The most famous example is the [halting problem](https://en.wikipedia.org/wiki/Halting_problem) which Alan Turing proved to be undecidable: Given a computer program, determine whether the program halts at some point, or runs forever. Turing proved that there exist programs where it is impossible to determine a correct answer.
And this was huge. Because it meant that it’s possible problems like the [Goldbach Conjecture](https://en.wikipedia.org/wiki/Goldbach%27s_conjecture), or the [Collatz Conjecture](https://en.wikipedia.org/wiki/Collatz_conjecture) are undecidable.
And with a little bit of work, using Turing’s result you can derive [Gödel’s results](https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems). Gödel proved that if mathematics is consistent, then it is incomplete: there will always be statements that are true but cannot be proven.
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