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This assertion requires an unambiguous definition for "bug" only then can we discuss the causes, origins and prevalence of them.
We have kernels like liquidSEL4 which is formally verified to have no bugs.
However, programs mostly rely on other programs ( your code -> your interpreter/compiler -> libraries -> OS -> firmware) and all of that relies on hardware.
Any of these can have bugs, there are hardware failures and poorly implemented or just worn down hardware.
It is almost entirely impractical for the vast majority of applications being written today, and the costs of developing perfectly bug-free software would outweigh the benefits.
A bug free program is possible unless your program has an infinite number of bugs. I don't think even IE6 has an infinite number of bugs.
It's impossible to protect against every possible situation.
Your code might work in normal circumstances or even be crash resistant but what if the computer explodes.
Just because something is possible does not mean it it probable.
The more complex the software the greater the likelihood of a bug.
so long as no new features are added or no new complexity is added to the software - given sufficient iterations of testing and bug-removal it may be possible to become bug-free?
Undesirable behavior will happen if our desires keep changing faster than software changes.
until bug getting found (cought?) a program is considered bug-free
Being able to recognize if an arbitrary program is bug-free would violate halting problem.
Therefore the statement can not be proved for arbitrary programs, which I assume, is the interesting case of the premise.
print(2 + 3)
This is a python program that adds 2 and 3 – and it has no any bug.