Why does Postgres handle NULLs inconsistently where unique constraints are involved?

Question

I recently noticed an inconsistency in how Postgres handles NULLs in columns with a unique constraint.

Consider a table of people:

create table People (
   pid  int  not null,
   name text not null,
   SSN  text unique,
 primary key (pid)
);

The SSN column should be kept unique. We can check that:

-- Add a row.
insert into People(pid, name, SSN)
values(0, 'Bob', '123');

-- Test the unique constraint.
insert into People(pid, name, SSN)
values(1, 'Carol', '123');

The second insert fails because it violates the unique constraint on SSN. So far, so good. But let's try a NULL:

insert into People(pid, name, SSN)
values(1, 'Carol', null);

That works.

select *    
from People;

0;"Bob";"123"
1;"Carol";"<NULL>"

A unique column will take a null. Interesting. How can Postgres assert that null is in any way unique, or not unique for that matter?

I wonder if I can add two rows with null in a unique column.

insert into People(pid, name, SSN)
values(2, 'Ted', null);

select *    
from People;

0;"Bob";"123"
1;"Carol";"<NULL>"
2;"Ted";"<NULL>"

Yes I can. Now there are two rows with NULL in the SSN column even though SSN is supposed to be unique.

The Postgres documentation says, For the purpose of a unique constraint, null values are not considered equal.

Okay. I can see the point of this. It's a nice subtlety in null-handling: By considering all NULLs in a unique-constrained column to be disjoint, we delay the unique constraint enforcement until there is an actual non-null value on which to base that enforcement.

That's pretty cool. But here's where Postgres loses me. If all NULLs in a unique-constrained column are not equal, as the documentation says, then we should see all of the nulls in a select distinct query.

select distinct SSN
from People;

"<NULL>"
"123"

Nope. There's only a single null there. It seems like Postgres has this wrong. But I wonder: Is there another explanation?

---

Edit:

The Postgres docs do specify that "Null values are considered equal in this comparison." in the section on SELECT DISTINCT. While I do not understand that notion, I'm glad it's spelled out in the docs.