Not wanting to put chemicals on the food I plan to ingest, I brought in bionatural, beneficial predators - ladybugs. That evening and the following morning I witnessed a decrease in the aphid population, as well as a decrease in the ladybug population. I have plenty of aphids to feed them, but they seem to be disappearing or dying. The plant under growth holds an abundance of dead predators, further exploration fails to reveal any young larva stage ladybugs. The aphids are back in full force.
My internet search comes up with a couple of other pesticide free solutions. The first is a tomato leaf spray, which is not supposed to harm beneficial bugs. So, I trimmed up the tomato plants, chopped the leaves and soaked them in some water overnight. Next morning I sprayed my plants and waited. That evening I was able to shake and brush off large amounts of aphids. I feel I might be making some progress.
The leafy greens I decided are a loss and I have pulled the plants - much to the chickens delight. I have come to the conclusion that the greens might do better outside of the greenhouse under some cloth.
The final option is the castile soap with garlic spray, this is a non-selective spray and will kill the good along with the bad. Hopefully, a few ladybugs will stay around and continue to decrease the aphid population allowing me to avoid the soap spray.