Naive Set versus Axiomatic Set Theories

Set theory is often viewed as the ``mother of all mathematics''. Much mathematics can be cleanly and axiomatically developed beginning with axiomatic set theory and then associating axiomatic rules to suitably defined sets and constructive relations. As suggested in the previous chapter, since the Laws of Thought sound a lot like statements in set theory and are the basis for the formal study of logic, it seems as though logic itself may be expressible in terms of suitable set relationships. The motivation for this is subtle. Historically, formal set theory came last, but the three form something of a tail-biting dragon. However, the symbolic language in which the laws of thought are expressed clearly came first, and already explicitly encoded an existential set theory that is the foundation of all human understanding as it is the basis of generalization and induction and ultimately, deduction.

I would therefore argue that set theory in actuality came first as it literally co-evolved with our generalizing human brains and a spoken language. All nouns essentially are symbols associated with sets of ``things'' that we group together on the basis of our perception or imagination of the real world, establishing an existential set theory and associated linguistic algebra with set-labelling words like ``tree''. It simultaneously developed a vast range of associated set selector and set transformation terms, e.g. adjectives, adverbs, and verbs.

Green trees take the set of all trees and extract the subset that are also ``green'' (where the adjective itself has a different meaning in reference to trees than it might in reference to a piece of paper, or a human face, or a piece of glass, or the wavelength of light). We can cut down the green trees, transforming a subset of all uncut green trees to a subset of all cut green trees, and further transform the trees to wood, to fire, to ash, to furniture, to dirt. It is important to remember that as this development of language was going on that permitted us to create a virtual image of a perceived external reality and symbolically manipulate it within the mental realm, that realm itself was also co-developing. Whole regions of the brain developed that are devoted to processing language and (especially) visual imagery, to automatically transform sensory input into neurological set representations. There is a strong temptation to get lost in the subject of how neuroscience, cognition, and language intertwine with evolution (something that is almost universally ignored by the mathematical philosophers that dominate the discussions of set theory) but we will bravely resist it^3.1

To fully understand the laws of thought, then, (which originally applied far more to trees than they did to arithmetic or geometry) we need to analyze their relationship to the existential theory of sets implicit in language itself before being formalized only a bit more than 100 years ago. We need to consider them from a semantic point of view as they really form an important part of the way our minds work without getting lost in a forest of contingent truths or assumptions - axioms.

At the same time, we need to keep the analysis consistent (more or less) with the formal development of set theory. The reason we are willing to settle for more or less is that even fairly careful set theory formulations are plagued with paradoxes^3.2 and antimonies^3.3 . When given a choice we will always elect a path that allows us to express the laws of thought in terms of sets in a way that most closely resembles their use in language and thought itself without worrying too much about formal symbolic logic or mathematics. As we shall see, this approach leads us to an existential view of naive set theory that is somewhat at odds with its more formal, axiomatic development but which clearly expresses the use of thought to analyze the real world, if not mathematics.

We adopt the point of view that no matter how you approach it, the development of mathematics requires axioms^3.4. In order to develop it from set theory or analyze it in terms of set theory it is not unreasonable for the set theory to acquire axioms which are then inherited by the mathematics. However, mathematics already existed and had axioms of its own before the invention of axiomatic set theory and retains some of these axioms even when developed from set theory. This has lead to some famous antimonies and a fair amount of (sometimes passionate, always entertaining) conflict.

However tempting it is for us to dive right in and join this fray, doing so would be a major distraction for us (and would consume half the book or more right there, as there is a lot of discussion on all sides of the issue, some of it overwhelmingly technical). We will therefore do our best to remain somewhat aloof from this debate by borrowing ideas from modern set theory without getting buried in its axioms or paradoxes.

Still, it will be difficult for is to do this (especially to do this without irritating mathematicians, mathematical philosophers, set theorists, and scientists) without some sort of review of the development of modern set theory, if only so that we can properly attribute ideas to their originators. We therefore will begin with a wikinote-dense short course in set theory. Or even a rather long course - Wikipedia's set theory offerings have more than doubled in number and depth since I began this project. We may well omit some critical reference or point of view or another in the process, but we will trust Wikipedia to ultimately balance this out with its rich set of crossreference links for those who really care to pursue it.

In that spirit, let us note that there are two general approaches to set theory^3.5 . The first is called ``naive set theory''^3.6 and is primarily due to Cantor^3.7 . The other is known as axiomatic set theory^3.8 or (in one of its primary axiomatic formulations) Zermelo-Fraenkel (ZFC) set theory^3.9 . These two approaches differ in a number of ways, but the most important one is that the naive theory doesn't have much by way of axioms.

We need two more results from existing set theory before proceeding. Both are associated with work done by Von Neumann^3.10 Von Neumann managed to create, for at least certain kinds of sets, a transfinite recursion of the generation of set objects that collectively are called a power set^3.11 that formed a Von Neumann Universe^3.12 .

The power set, in particular the set of all subsets of an actual existential Universe of objects, will be a key element of our discussion of the laws of thought. We will keep all subsequent discussion of the laws of thought carefully grounded in reality in this way, and only later will we worry about paradoxes and formal developments in axiomatic set theory and number theory and the like. As we'll see, there are tremendous advantages to be obtained from doing so. For one, we can keep the whole discussion ``naive'', indeed naive at the pre-Cantor non-axiomatic level. For another, we'll see that in an existential set theory of this sort, perhaps unsurprisingly, paradoxes cannot happen. At least we hope so. That's the whole point of the laws of thought, after all.