Why can no program solve the halting problem for every possible program?

The halting problem is a fundamental concept in computer science that demonstrates the inherent limits of computation. It asks whether it is possible to create a universal algorithm that can determine, for any arbitrary computer program and input, whether that program will eventually stop (halt) or continue to run forever.12

The reason no such program can exist is rooted in the work of Alan Turing. By modeling computation through the Turing machine, Turing proved that there is no general-purpose method to solve this problem for every possible case. If one were to construct a program capable of deciding the halting status of all programs, it would lead to a logical contradiction—essentially creating a scenario where the program must halt if and only if it does not halt.12

This result confirms that there are undecidable propositions within formal systems, meaning some problems are mathematically impossible to solve algorithmically.12