Set Theory of Thought

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In this introduction so far you've already seen at least glimpses of many of the basic punch lines of this book, but they're probably a bit amorphous yet (at least I hope that they are or you won't keep reading). Either way, we covered a lot of ground so let's summarize what we learned before moving on.

At this point we should be able to see that set theory is all really
lovely and seems somehow to be more fundamental than the rules of logic
and mathematics or the Laws of Thought. We've also seen how it appears
possible to make a naive, existential set theory that eliminates the
possibility of paradox while embodying the Laws of Thought in a way that
at least seems *less* ambiguous than they did in English.

In the process we deduced some important truths about the necessity for
matching the *domains* of proposed set relationships intended to
pick out particular sets from the set of all sets within our existential
set Universe (which are all there whether or not we pick them out).
Since we can easily come up with *silly* relationships, or broken
relationships, or paradoxical relationships, or self-referential set
relationships that do *not* describe a set in the set Universe
(including even the empty set) we invented a ``set'' that is *not* a set, the NaS set. A metaphor for this set (which is *only a
metaphor* since it isn't a set and doesn't exist in the closed set
Universe where set operations are defined) is that it can be thought of
as the *non*-invertible complement of the Universal set we wish to
reason in.

At this point, formal logic is one possible thing that can be built on
top of or in parallel with existential set theory - we just add a very
few axioms and definitions and stir gently, since the Laws of Thought
(viewed as axioms or not) are built right into its basic operational
structure. We do need to discuss and define the notions of ``true'' and
``false'' and how they differ from ``exist'' and ``don't exist'' (are
null) in the set theory, discuss the notion of ``provability'' as a
possible proxy for ``is true'', and so on (and will do so in the next
chapter), as those things appear to be *algebraic* constructs that
gain existential validity only to the extent that they permit us to make
*well-formed propositions* concerning their associated Universal
set. In general we'll find that they are really useful only in *artificial* set Universes, not existential ones, and only useful to
existential ones to the extent that we construct axiomatically defined
mappings between the two. The real Universe isn't ``true'' - it just
is. Abstract propositions in logic about the real Universe can never be
*proven* to be true using logic without this *presumed*
(unprovable) mapping.

Formal mathematics is what we call the human activity of creating the
artificial set Universes we wish to either use as a proxy for the real
existential Universe or just for the sheer fun of doing so. It is
usually developed from *axiomatic* set theory from the beginning
because we can see almost immediately upon attempting to develop set
theoretic mathematics that any such development cannot be unique or
complete. Axioms are needed almost immediately for non-existential
mathematical sets to deal with notions of conditionally undefined set
operations, paradoxes, domain restrictions and infinity, and oddnesses
that result from viewing any given Universal set - say, the integers -
as being embedded in a larger Universal set - say, a quaternionic field
expressed as a function of curvilinear space-time coordinates with
specific conditions on smoothness and a metric. Cantor's paradox
(really, Cantor's theorem) suggests that pretty much any set Universe
can not only be embedded in a larger set Universe, it can be embedded in
a *much much larger* set Universe, recursively, so we can never talk
about a *truly* Universal Set that contains all possible Sets any
more than we can talk about a largest real number, only various ways
that real number sequences can scale to infinity.

Computational mathematics is a particularly lovely blend of logic and
arithmetical mathematics on a highly constrained, discretized domain.
The ``set Universe'' of objects acted on by the operations of a computer
is finite and discrete, intended to *approximate* real number
arithmetic via integer arithmetic and symbolic mappings on a finite
mesh. As a result it is nearly ideal for our purpose of understanding
the (NaS) set, especially since there is of course no way to *actually* produce a truly undefined result in the limited set of logical
transformations available to a computer.

Physics and natural science in general are still another and are *truly* based on an existential set Universe, even though without axioms
even there we find ourselves unable to reason about the set Universe,
only to experience a single instantaneous realization drawn from it. We
*imagine* that there is something we have *named* ``set theory''
(and all related imagined results such as ``logic'' and ``mathematics'')
but to be able to *use* this to *reason* about the actual
existential set requires axioms galore.

Just as is the case with computing, where NaN results have to be encoded
``by hand'' based on higher order definitions and axioms ``inherited''
from a presumed mapping to a particular space expressed in terms of
discrete binary transformations so (NaS) results can only be
encoded for our human brain ``computers'' as metaphors, higher order
axiomatic construction within the confines of the existential set
available to our sensory perceptions. In neither case do our computers
or our brains actually generate a ``set'' (result) that is not in the
theory - rather they agree to *waste* one or more of the sets in
the set Universe by assigning them the *meta-meaning* that results
mapped there are not ``results'' and do not belong in the set Universe
in question.

is as inconceivable to human reason as ``infinity'' is to a
discrete binary computer, yet we can build a computer program that
identifies arithmetical operations that would produce infinity or
undefineable results and cause those operations to be consistently
treated according to a table of operational results that determine what
one gets when one e.g. adds ``infinity'' to a finite number (get
infinity), divide a finite number by infinity (perhaps we wish to make
this zero), subtract infinity from infinity (usually considered to be an
undefined result). Truthfully, the only way the human brain itself
understands infinity is in terms of tabular constructs just like this -
we understand infinity in terms of how it behaves as the result of an
*imagined* limiting process. In ordinary arithmetic is
undefined, yet in physics
is used all the
time because *for all nonzero no matter how close to zero one
gets* the result is well-defined^{3.42}.

In all thes cases, in order to *reason* about the probably infinite
number of subsets of the presumed Universal or non-Universal set, the
very general framework of an abstract set theory is clothed in *imaginary clothes* - unprovable *assumptions* called (in various
contexts) definitions, axioms, rules of inference, laws of nature,
microcode of the computer, a language (with a dictionary of symbols, a
semantic mapping of those symbols, a syntax one can use to assemble
valid ``statements'' with the language, and more). These are all rules
that exist in our *minds* as we *reason* (or that appear to
exist by *inference* in nature as it operates or that are engineered
into the computer under the same assumed inference rules as those of
nature or mathematics but applied). These abstract, arbitrary rules
both select particular subsets of the existential Universal set via *metaphorical* relations established between that set and particular
abstract mathematical sets and identifies them (encodes them within a
language) via *information compression*, with inherited axiomatic
functional relations between them. This is how we reason.

Since assumptions - specifically the *axioms* of a theory - seem
to play a pivotal part in where we go from here (and is, after all, the
title of this book) we'll next *explore* axioms in the context of
logic and try to better understand just what they are and how they occur
throughout *every realm of human endeavor* as a prior step to being
able to do anything ``rational'' within those realms at all. In the
process of looking at axioms and their critical role in formulating any
sort of system of reason, we will inevitably be drawn to look at what
can only be called a kind of breakdown in the scope of reason itself -
Gödel's Theorem.

This is basically an extension of work done by Russell himself, in
particular, the Russell Paradox. Although its importance *can* be
overemphasized - it certainly doesn't mean that reasoning itself ``no
longer works'', it doesn't mean that mathematics and physics and all
that are in any fundamental sense unsound - its importance should not
be *minimized*, either. It definitely warrants its own dedicated
discussion, as it *both* maps certain paradoxical logical constructs
out of set theory and into the null set *and* separates the notion
of ``truth'' from that of ``provability''.

This is quite disturbing. It turns out that we cannot even completely
analyze *mathematical* or *logical* propositions for truth
within any sufficiently complex axiomatic system. As noted in the
somewhat irreverent beginning of this part of the book, we want answers
to the Big Questions - the questions on Life, the Universe, and
Everything. We've assigned the task to Philosophers and want them to be
able to explain their answers to us ``beyond any doubt'' using this
reason thing we've been paying them to work out for so long. And it
isn't just everyday people that expect pure reason to pay its own way
and explain it all - even relatively contemporary and quite competent
philosophers-mathematicians like Bertrand Russell would like
``startling'' conclusions to come out of the simple rules of logic
applied to the world^{3.43}.

Alas, as we will see in great detail, we will be forced to conclude that
Russell's fond hope is doomed from the beginning. We cannot prove *one single damn thing* about the Universe using *pure* reason. To
use reason at all we must begin with unprovable axioms (unprovable
premises leading to unprovable conclusions according to logic itself)
and when we do, we will *almost certainly* still be left with some
things that may be true or false or both or neither but unprovable in
any event.

Now let's get started. It is pretty clear that our examination of
``reason'' must continue with a look at the *details* of how logical
systems work. We've laid a good foundation with our discussion of set
theory above, and I've laid out some of the things we hope to learn
already so they won't come as a complete surprise or shock, but the *details matter*. Part of the ``stomping the corpse'' thing.

Logic

Set Theory of Thought

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