You should definitely read up on the halting problem.
Yes, it can, that's what the whole formally verified software branch is about. Of course, even with formally verified software, there can be failures caused by external factors such as faults in hardware etc. But some piece of software itself absolutely can be proved to be correct (ie. bug-free). It just takes special approaches such as programming languages amenable to formal proofs (eg. Idris).
Software is usually written in bug-inducing ways and using languages that hinder formal verification not because provably bug-free software is impossible but because it is typically too costly and difficult to write.
Of course this still leaves space for bugs outside the program itself which still may influence it, such as bugs in the operating system or firmware or the like. Which may be prevented by having a formally verified OS too :)
But even in such a case, there might still be hardware errors (electrical noise, for example). Which is why for example spacecrafts or critical industry machinery double or triple their whole architectures. In such case the formally verified software runs on eg. three instances simulatenously and these instances cross-check each other's results all the time. When a hardware fault occurs (a bit randomly flipped in memory for example), they find out they're not matching and re-calculate.
That's as close as you can get to zero bugs.
Sure you can: you can prove software is correct through exhaustive testing. As noted previously it only works for small input space and simple programs (functions, really) but it does work, you can prove that a boolean xor is correct by enumerating its 4 inputs and checking that all of them produce the expected output.
This method can be applied to input spaces up to about 40 bits or so: https://randomascii.wordpress.com/2014/01/27/theres-only-fou...
What an unexpected coincidence, so am I!
> It's unlikely that anyone cares if a Hello World program is bug-free.
Did you consider reading the link I provided at any point?
You can't "stick by your assertion" then provide something which has almost no relation to your original assertion, that's called "moving the goalposts".
This moving of goalposts is even more asinine when you're moving them exactly where the comment which you originally decried had put them.
Reminder: here is what you originally felt you needed to note was wrong:
> Software can not be proved bug-free by tests (and even that assertion is not completely true, you can prove software through exhaustive testing if it's very very simple).