Cautious virus detection in the extreme

It is well known that there exist viruses whose set of infected programs is undecidable. If a virus detector is to err on the side of caution with respect to such a virus, then it must label some perfectly innocent programs as being infected by the virus. Can there exist a virus whose set of infected programs is so unwieldy that any cautious virus detector must label all but finitely many programs as being infected by the virus â even when infinitely many programs are not infected by the virus? Although such viruses can exist, strong theoretical evidence is presented that such a virus is unlikely to be encountered in the real world. Several of our proofs employ infinitary self-reference arguments