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-defined3.42.
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 world3.43.
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.