The idea is indeed logical and stupidly obvious, once you learn the basics of what "optimization" means, or what "recursion" is.
> I find that hard to believe. Ever watched Terminator?
Terminator has fuck all to do with recursive self-improvement. Don't confuse people who grew up on sci-fi with people who casually went to see Terminator or some other pop-culture artifact featuring some kind of "AI".
> OK, then. You're a GI, go off and build an army of better yous and take over the world.
What do you think the drama with eugenics, genetic engineering and designer babies is around? It's literally humans trying to make better humans in the only way that is available - reproduction.
AI made in silica would be more malleable, easier and cheaper to replicate. Self-improving software isn't even a fantasy; it exists in many forms - though it's far from open-ended like a self-improving GI would be.