Formal Set Theory

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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.

The Power Set

Formal Set Theory

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