Fortunately, it's getting harder without previous addresses or other verification methods.
For non-Americans that don't know, our Social Security number is generally assigned at birth or when you become a citizen by the Social Security Administration. Social Security is a disabled or elderly benefit we all pay into (roughly 7.5% employee and 7.5% employer - ~15% total). It's the only number we all get, since not everyone gets a driver's license; ID; passport; or other identifier. Unfortunately, it's been used to identify us for everything, and until recently was typically in plaintext on most forms (medical; tax; student; etc...).
CGP Grey has a good summary of how it came about and why it's become a problem: https://www.youtube.com/watch?v=Erp8IAUouus