Wow, not to put too fine a point on it, but your A/C guy sucks.
Our furnace gave out, only in the 30-40 range here, and our guy had it fixed the next day... including a special trip to get a specific part because our POS old and cheap system can't work with the 'universal' parts that he had with him. These guys (and also my Uncles both in A/C in different companies) do NOT mess around when it comes to a family freezing or overheating in summer either. They do their job and they do it fast. Seems crazy to me that you've been freezing for three days. Is there any other A/C company in the area you could possibly call? Seems so weird to me. When our system crashed I called Uncle and he walked me through the troubleshooting, easy peasy it's the ignitor. Not that I had one of those lying around, but I felt better knowing it was a simple thing you know? When we first moved in, first time it got cold we tried turning on the furnace and... NADA. Nothing. Zilch. So called Uncle and he walked me through trouble shooting. Nothing found. So 'we' took apart the thermostat and lo and behold someone had removed the copper wire that (basically) connects the switch to the furnace... without that wire you can flip the switch all day long but the command never makes it to the furnace so no heat. I'm guessing it was our dear landlord, securing the house while empty, and forgetting to let us in on the joke... but we got it fixed, I stripped some wire from another source, wrapped around the posts, and done. Pop the face back on the thermo and we're good to go. Didn't have to freeze overnight, didn't have to call the landlord and hope for an A/C appointment. Yuppers, very nice having uncles in A/C, even if they are too far away to come running to help.
So anywho, that's why I'm wondering how it's possible that this guy hasn't figured it out yet. I mean, there are basic troubleshooting steps (assuming you've been trained that is) that lead you to the problem. Shouldn't take more than a couple hours MAX to run through them... so how's he still bumfuzzled after three days? That's making me very nervous.