A knockdown prophecy

[The Grand Nudger]

This is devolving into a cheap game of semantics. Even if I play along, I can still squeeze a first modern human from your description above, regardless of its accuracy.

Let's assume we observe the first generation of modern men dispersed over groups in different places around the globe.

Step 1 : For each group of "modern men", take the oldest member.

The oldest member even in this thought experiment would be an archaic, not early modern, or fully modern, human.  

Step 2 : You get a collection of very old people, each of whom would locally be considered the first modern human.

So very many adams....

Step 3 : Take the oldest element in this collection. This element would evidently be the first modern human we're looking for.

This one is just..bonkers..and I think you can figure out why it won't give you an adam all by yourself with reference to living families.  Picking the oldest member of our two families...will not yield a common ancestor between you and I.  

QED.

The existence of a greatest element is a mathematical property about finite sets, the set of modern humans is finite, people.

Yes, QED, you have proven yourself wrong.