If someone tasks me to create a set of even/prime/blue/rectangular/crunchy/uninteresting numbers I have two options:

1) I list each and every number that is part of the set. It is OK if the set is countably infinite, we can wait.

2a) I grab my special black box that receives a number and lights up a red or a green LED depending on whether the input is a member of my conjured up set or not;

2b) I grab the other special black box, this one has a single LED (to indicate it is switched on) and a push button which prints out the next member of the set on infinite 7-segment displays. The box is a bit wider than the 2a) unit.

These are mostly traversable, e.g. my 2b) generator could be built from a counter and a 2a) tester, or my 2a) tester could use a table lookup backed by a 1) list for all I know.

What they can/should not do is retroactively change their mind on the membership of a particular number:

- It is either in the 1) list or not, no erasers, no backsies;

- 2a) should always respond with the same LED for a given number, no moon phase lookups, no RNG, no checking of previous LED responses;

- 2b) can not even be rewound so it is impossible to tell if it would produce or skip the number, should we coerce it somehow to start again (we can't).

So using any of the two and a half mechanisms lead us to a set where the minimal element should have the same property as any other element: it is exactly as even/prime/blue/rectangular/crunchy or uninteresting as the rest of the set.