I suppose there is room for confusion here because some regulator IC's provide a constant current instead of a constant voltage as the reference.
For example, the LT3080 that Dave is using in his power supply can be run in one of two modes. It regulates the output to be equal to the voltage on the set pin, so if you use an external voltage reference to provide 5V to the Set pin, the output will be 5V. Using it in this way, the LT3080 acts as a regulator without an internal reference.
However, if you connect a single resistor from the Set pin to ground, the LT3080 does become a fixed voltage regulator. What is going on? You cannot use the 0V Set to Output differential as a voltage reference (as you can with the LM317) , so how is it possible for the LT3080 to supply a fixed output voltage without an external reference?
The answer is instead of providing a fixed voltage (like 1.25V form the LM317), it provides a constant current from the Set pin instead (10uA). Thanks to Ohm's Law a constant current into a resistor gives a constant voltage.
This is not the only regulator using a current source output to generate the voltage reference - there are lots of other linear and switching regulators using the same strategy. There is every chance that the LT3080 internally has a voltage reference to generate the constant current out of the Set pin, but it is possible to make constant current devices in silicon without any voltage reference involved.
So basically to get a fixed voltage out of a regulator without any external reference voltage, then a regulator IC must provide either a constant voltage reference or a constant current reference, and thanks to Ohm's Law, either one is equally good. With a single resistor a constant voltage can be turned into a precise current, and a constant current can be turned into a precise voltage.
As IanB explained so well, without some sort of fixed reference, there is just nothing to measure the output voltage against.
Richard.