Not to Invite "Extras" to the Wedding
It is OK to tell those who are coming that they cannot bring a date. It is your wedding and as such you have the ultimate control over who should attend. I wouldn't want to have 15 people I know show up and they in turn bring 15 people I don't know. These are people you are paying for their meal, drinks, and entertainment for the night.
For my wedding we actually had just family there and it was an extremely small wedding consisting of my mom, aunt, uncle, twin sister, brother in-law, my niece, my husband's mom, the preacher, and the preacher's wife. We also got married with short notice of two weeks to everyone in order for them to be there. In our defense we planned the wedding for June but had to postpone it until August when I returned from my deployment.
My twin sister's wedding had over 100 guests and most were from our church and my brother in-law's family. We did a pot luck for her wedding which kept the food cost down so money wasn't so much of a factor for her wedding.
The questions to ask yourself are:
1. Do you want strangers at your wedding?
2. Are you limiting how many people are attending?
3. Is money or space a factor of limiting the size of your wedding (if #2 is yes)?
Once you have the answer to the questions above, you can evaluate if you are going to tell your guests they can bring guests themselves or be "single" for the wedding.