If you ripen papaya on the tree in my part of the world the birds would get it first. Or it would fall like H. Dumpty and all the king's horses...
We just leave them on the kitchen table and they ripen.

Papaya trees, BTW, are male, female or hermaphrodite. Just like many of today's humans.

However, I would stay clear of papaya, L Hawk. It contains papain which is a meat tenderiser. If you have an old cock who's been with you for many years and he gets papain on him, he'll soften up pronto.

This is often not a desirable state of affairs.