Bob would have to know and trust Grace beforehand. Grace could be the IRS, for example. The idea here being to have somebody who already knows your age vouch for your age.
Here's my idea: Bob gives Alice a token, assigning her an unique random number n. Alice goes to Grace and tells her, "Somebody assigned me number n, can you verify that I'm allowed?" Grace then writes: "User n is allowed, signed Grace". Alice then takes this letter and shows it to Bob. Bob now knows that Alice is allowed, but nothing else. Grace only knows that somebody wanted to know that Alice is allowed, not who that somebody is.
Of note here: This system does nothing to protect against an allowed user helping a not allowed user to gain access, but I don't think it's possible to protect against traitorous users.
Yeah, that is a problem.
And thanks for the compliment.