This sequence is obtained by iterating a functor that creates a new set from the union of the preceding two sets, thus generating sets with the cardinalities 0, 1, 2, 3, 4, ad infinitum. In less mathematical terms, the principle can be described as follows: Beginning with emptiness (step 0), we observe emptiness. Through the act of observing we create an entity containing emptiness (step 1). Now we perceive emptiness, as well as an entity. From the combination of the former two we create another entity by observation, which is different from the first entity (step 2). This process is repeated again and again. Interestingly, if we define suitable operations on the obtained sets based on union and intersection, the cardinalities of the resulting sets behave just like natural numbers being added and subtracted. The sequence is therefore isomorphic to the natural numbers - a stunningly beautiful example of something from nothing.

 

 

Step 0: { } (empty set)

Step 1: { { } } (set containing the empty set)

Step 2: { { }, { { } } } (set containing previous two sets)

Step 3: { { }, { { } } , { { }, { { } } } } (set containing previous three sets)

Step 4: { { }, { { } } , { { }, { { } } }, { { }, { { } } , { { }, { { } } } } } (etc.)