Specification
http://docs.oracle.com/javase/8/docs/api/java/util/Properties.html#stringPropertyNames--
says:

"The returned set is not backed by the Properties object. Changes to this Properties are not reflected in the set, or vice versa."

The "vice versa" part assumes that the returned set could be changed. In fact the returned instance throws UnsupportedOperationException for an attempt to add anything. The specification needs to be updated/clarified.

Please see the following code:

public class PropNames {
    public static void main(String[] args) {
        new java.util.Properties().stringPropertyNames().add("abc");
    }
}


The output will be

Exception in thread "main" java.lang.UnsupportedOperationException
at java.util.AbstractCollection.add(AbstractCollection.java:262)
at java.util.Collections$SynchronizedCollection.add(Collections.java:2035)

This doesn't look perfectly complying to the assertions from the spec

The following JCK9 test based on the current spec will fail due to this:
api/java_util/Properties/autd2.html#StringPropertyNames[changesAreNotReflectedInTheProperties]