In Integral Relational Logic, set is the primal concept of interpretation, turning meaningless data structures or graphs of relationships between forms or elements into meaningful information and knowledge through the fundamental principle of concept formation.

Sets are formed by carefully examining similarities and differences in the attributes of beings, whose members are entities, as instances of classes.

There is thus no need for the complexity of the axioms of set theory in mathematics. We just need the commonsensical definition of set, as Georg Cantor conceived it: “By a set we mean the joining into a single whole of objects which are clearly distinguishable by our intuition or thought.”

As the elements of a set can be anything whatsoever, they can be sets themselves. We can thus envisage a largest set, defined as the ‘set of all sets’. However, for any set of n members, the set of all subsets of a set has 2ⁿ members, even when n is infinite, leading to an infinity of infinities without end. So, the ‘largest set’ is the largest, by definition, and not the largest.

This paradoxical situation, and others like it, doesn’t matter in Integral Relational Logic, for its sole axiom is the Cosmic Equation, as the fundamental law of the Universe. At this stage in the development of this universal art and science of reason mathematics does not yet exist.

It is first necessary to demolish this vast edifice so that we can rebuild it on a fresh foundation. In this way we can see that mathematics is the abstract art and science of patterns and relationships that emerges directly from the Divine Origin of the Universe in the vertical dimension of time, like all other structures in the relativistic world of form.

As this concept of set is so simple and fundamental for all our learning, not just in mathematics, in the 1960s, some mathematicians on both sides the Atlantic set out to teach sets to eight- to eleven-year-olds in primary and elementary schools. As the authors of The ‘New’ Maths pointed out, the new maths was intended to bring meaning to mathematics and hence to all other disciplines. But it seems that this initiative was abandoned because children were not developing the numeracy skills required by business and science, inhibiting evolution from becoming fully conscious of itself within us human beings.

Nevertheless, here is a basic example. As children we learn to distinguish red, green, and blue blocks and the shapes of circles, triangles, and squares, putting them into groups of blocks with similar characteristics. We can see here that the number of elements in these sets is either two or three. So, we can form sets of sets with a specific size, showing that the concept of number is dependent on that of set.

Nevertheless, some adults are aware of Venn diagrams, depicting the logical relationships between sets with various attributes. Here is an example, illustrating a universe of discourse of all English words, some of which have the same meaning, spelling, or pronunciation, or combinations of these, as synonyms, homographs, homophones, heterophones, heterographs, and homonyms. The white space surrounding this diagram is that part of the universal set that contains those words that do not have such similarities.

However, Venn diagrams are not much used in the foundations of Integral Relational Logic. The relationships between relations, as used in database design, are a far more powerful way of representing the underlying, unifying structure of the Cosmos.