Can we have an antivirus that detects all viruses?
No math needed. Experiment time. Lets arrange a pool of viruses, a pool of systems-that-try-to-defend-themselves and mechanisms that constantly change/improve both parties.
Give them a planet. Wait billions of years.
Do we still have viruses?
For me the sheer number of possibilities already tested is far beyond satisfactory.
About "proving"
Here I'd like to explain why I've drastically reframed your question, which was originally about - lets simplify a bit - people writing about mathematics:
You can only prove anything in axiomatic systems ("worlds" that are ruled by a set of axioms - where axioms themselves are defined as unprovable). I don't know axioms that rule the real computer viruses, or real biological viruses, so I cannot prove anything about these.
I'll try to predict reality and I'll be often correct, sure. But I will use falsifiable theories for that, not proofs.