We can't agree on what can or cannot be computed unless we agree on a reasonable model of a computer. The most basic computer is a mathematical concept called a Turing machine .
Counter programs are a very different, yet equally powerful, computational model.
The Church-Turing thesis states that all known models of computation are equivalent. Using these formal models of computation we can show that there are certain problems that are undecidable , which means they cannot be solved by any computer or computational model.