A table in a relational database isn't really a table unless it has a primary key; there <i>must</i> be a way to uniquely identify each row, otherwise, for example, how would you write a SQL statement to update a value in a specific row? Consider reading a relational database primer, eg
http://www.artfulsoftware.com/mysqlbook/sampler/mysqled1ch01.html.
Auto_increment is a mechanism designed to do just one thing: create keys guaranteed to be unique. Auto_increment keys are
surrogate keys. You can form primary keys from one or more existing columns (
candidate keys).
US SSNs, however, are poor candidate keys. Why? Generally, if a key depends on the outside world, it is subject to change, but primary keys shouldn't require edits. SSNs are often fake, often misreported, and some people don't have them or don't know theirs. In addition, some states have laws restricting the use and display of SSNs. Don't try to use SSNs as PKs.