Sets and specific differences

 

Brouwer’s understanding of language uniquely on human terms justifies his separation between language and mathematics but does not make any psycho-biological sense. The act of distinction which characterizes the intuition of time in Brouwer’s sense precedes the use of our human languages but not of languages in general, i.e., of communication among living creatures. A twoity, therefore, precedes also the linguistic construction that we call number -even the concept of numerosity- and denotes an action of separation. For this reason I will call merisma (part), to such an action of distinction. A merisma  is a precondition for any action, no matter whether such distinction is accompanied by second-order distinctions in relation to the cognitive actor. A merisma has the irreversible character of the intuition of time, of life experience, and therefore, implies a fundamental or primitive determination for later distinctions. We can notate this primitive merisma by ‘≠’.

The communicative expression which iterates the primitive merisma is a primitive sequence. When we cannot apply ‘≠’ among two instances (parts) of the iteration, we say that those instances are equal. The double composition of  ‘≠’ and ‘=’ over a primitive sequence could be called fundamental morphism. A fundamental morphism is therefore, an action of identification. Two fundamental morphisms are different when we can express a merisma with their composition: such distinctiveness is called a specific difference. 

A set is an algorithm (procedure) for the construction of a specific difference. We could also understand it as a finite semantical action for the synthetic construction of a specific difference. The synthetic representation of the algorithm under the concept of set adds an epistemological content of totality which is not necessarily implied in the constructive process, and which, in fact, hypostasizes a transcendental property. An object ‘s’ belongs to a set S, notated s ∈ S, when the set generator algorithm, or inclusion map, can be performed with ‘s’. The void subset is a rather serious ontological problem. Constructivist definitions, like Bishop’s, appeal to the idea of a set whose construction produces a contradiction, or whose inclusion map is identical to the inclusion map of its set. But a contradictory construction is not an algorithm, therefore, there is not such a thing as an empty set. However, we can produce an object called empty set as the result of the operation of extracting one element in a subset of precisely one element. But what is the meaning of such set? It is an operational structure which simplifies the manipulation of sets. 

A relation is an algorithm for the construction of a specific difference pairing two given sets.

We can speak of specific differences of specific differences, and so on, but we do not have a set unless we have an algorithm for their construction, unless we have inclusion maps. The postulate of an algorithm that generates any other algorithm is equivalent to a postulate of omniscience, inconsistent with our psychobiological conditionings and constructively meaningless. This postulate implies that the axiom of choice would be meaningless.