Can the equality of two real numbers always be determined?
Let us say that we have derived an expression for a real number X. We also have obtained an (entirely different) expression for a real number Y.
Now one mathematician claims that X and Y must necessarily be exactly equal. Another mathematician claims that X and Y can not be equal, however their difference is extremely small; say 0.5^N with N some unknown but very large number.
To settle the dispute the mathematicians decide to use a Turing machine.
As input the machine takes the binary expansions x(n) for X [n=1,2,3,4...] and y(n) for Y. If the machine finds that for some n the pair of digits x(n) and y(n) are unequal, then it concludes that X and Y are unequal and the program halts. However if X and Y are equal, the machine will continue to check digits "ad infinitum" without ever reaching a conclusion. So in the case of equality (X = Y) there appears to be no halting criterium.
In the example that I described, the Turing machine is of no great help. The machine just continues to run "forever". Thus it remains unresolved whether this is because X and Y are exactly equal, or whether they are unequal but the first difference in their digits occurs at some unattainable high index N (required CPU time > age of the universe).