Is the catalog file marked as read-only?
No. When I click on "Get Info" for the file it says that I can read and write to the file under "Sharing and Permissions". Is there somewhere else I should be looking?
I'm not a Mac person, I don't know where else you should look, but maybe someone else (or Google) can tell you.
What do you mean "I have checked the file permissions"? Did you check the permissions of the folder that your Lightroom catalog is stored in?
Also, What file? Are you sure this is the exact catalog file Lightroom is trying to open? What happens if you double-click on this catalog file that you claim has the proper permissions?
I am opening the .lrcat file. When I open it, I get the message I described above: "Lightroom cannot use the catalog named 'Lightroom 3 Catalog' because it is not writable and cannot be opened. This could be caused by incorrect permissions or because another Lightroom application is using the catalog. You may try to the correct the problem or you may select a different catalog.'' This happens if I open it directly from the finder or from within Lightroom.
When I google this issue all the advice I see is to delete the lrcat.lock file and I have already tried that and it is still not working.
When I say I checked the permissions, I mean I clicked on "Get Info" for both the file and the folder is is in and both list me (as in, my user account) as being permitted to read and write to the file and folder. These are the same permissions that I have on a different, functional Lightroom catalog on a different machine. When I googled the issue, I could not find any answer about what could be wrong with my permissions (according to everything I have found they are set correctly). Hence my frustration.
You haven't addressed the issue of whether or not you are looking at the permissions on the folder/file that Lightroom is actually using. How do you know that Lightroom is not trying to open some other catalog? You need to nail this down with 100% certainty.
You haven't tried double-clicking on the catalog file of interest to see what happens.
When I mentioned Google, it is because I am not a MAC person, but the issue you need to look up is how to make sure the catalog file is not set to be read-only in your operating system. On Windows, permissions is different than setting a file to be read-only. On Windows, you can have write permission, but if the file is read-only, you still can't write to it.
I am in fact double clicking on the lrcat file. I get the error message when I do that. I get the same error message when I also try to access that lrcat file from with Lightroom. Either way, same message.