Functionalism and pain-likeness

Say that a functional property F is pain-like provided that a human is in pain if and only if the human has F.

Assuming functionalism, there is a functional property F₀ which is pain. Property F₀ will be pain-like, but it won’t be the only pain-like property. For there will be infinitely many ways of tweaking F₀ to generate functional properties F₁, F₂, ... that in humans are instantiated precisely when F₀ is, but that differ in instantiation among aliens. For instance, F₁ could be F₀ conjoined with the property of not currently thinking a thought that has seventeen levels of embedding (I take it that humans can’t think a thought with more than about three levels of embedding), while F₂ could be F₀ conjoined with the property of not consciously exercising magnetic sense, and so on.

There will thus be infinitely many pain-like properties that differ in when different aliens instantiate them. One of these pain-like properties, F₀, is pain. And now we have a difficult question for functionalism: What grounds the fact that this particular pain-like property is pain? Why is it that having F₀ is necessary and sufficient for hurting but having F₁ isn’t? What’s so special about F₀? Why is it that F₀ picks out a phenomenally unified type, but the other properties need not?