Paradoxes

Other Books by rgb: |

The need for a null set isn't something I'm just making up. No
matter what you *call* it, it has been around for a very long time
in mathematics and logic. It has been around for a somewhat shorter
time in computer languages, because computers themselves are relatively
new, but (as pointed out by Jaynes) a ``robot'' (or computer) is an
excellent model for logic because of the entirely *practical*
constraints associated with *engineering* it to actually work.

Computers are nearly ideal embodiments of the raw *mechanics* of
logic and arithmetic. Computers more or less directly implement a
system of boolean logic as the underlying basis for their operation -
all successful operations have an associated ``truth table'' and can be
described by a set of permissble binary transformations with the
representation. However, the representation itself is *finite and
discrete* and can be *self-referential* - one can program a
computer to program a computer and work other dark magic (some of which
is covered from time to time by such luminaries as Martin Gardner in
*Scientific American*).

Computer programmers typically learn in their very first class - the
hard way - about the null set as the result of certain
operations. Computers can easily be asked to take perfectly ordinary
and reasonable results as input to operations that produce *no*
meaningful output. That is, they have to deal with ``Not a Number''
(NaN) return conditions in addition to *other* (possibly domain
specific) undefined or oscillatory conditions *all the time*.

For example, there is nothing to prevent *algebraically* well formed
computer programs from attempting to divide by zero or take the inverse
sine of 2.0, things that are *algebraically* sensible but that
happen to be undefined for particular values or ranges of inputs.
Indeed, it happens all the time, as people write a program without
checking and the program manages to generate a forbidden value as input.
Similarly it is easy to write a program that enters a loop such that the
value that is tested to determine when to terminate the loop oscillates
and hence never reaches the end condition with a well defined answer.
One might even write such a program deliberately, if the program has
some desired output that is incidentally generated along the way, and
terminate the program only by killing it from ``outside the Universe''
of the program, with a Ctrl-C from the keyboard.

Once a NaN or indeterminate state is introduced into any computation by any means, all operations that include it in any way become equally undefined and result in NaN as well. In fact, I was very tempted to use NaE (Not an Element - of a set) instead of in the formulation above to draw careful attention to the correspondance.

Nor is the abstraction of the null set unknown in formal logic. is the essential statement of contradiction^{7.1}^{7.2}

If you ring a contradiction into a theory (however subtly) you can prove
anything you like *symbolically*^{7.3}. The argument goes
something like this:

- Let us state as an absolute truth some : ``Robert Brown is a wise fool'' (where we define ``wise'' to be the same as ``not a fool''). This is a fairly familiar oxymoron in English, and is probably even true...
- Suppose that God is Not A Penguin (GiNAP). Seems like a perfectly
plausible assertion, at least in the minds of some, although it does
somewhat beg the question of whether or not God exists to
*be*a penguin. Fortunately, it is only an assertion. - Well, if Robert Brown is wise (and he is, see the first line),
then it is certainly true that ``if GiNAP, then Robert Brown is wise''
(and, for that matter true that ``if God
*is*a Penguin, then Robert Brown is wise'' - you begin to see the trap). - This is
*logically equivalent*to the statement ``if Robert Brown is a fool, then God*is*a Penguin''. - But Robert Brown
*is*a fool - see the first statement! Therefore... - God is a Penguin!

We've just proven that God exists *and* is a Penguin^{7.4} all in one
simple derivation^{7.5}! Bertrand Russell would be so
proud...^{7.6}

Or rather, we've proven no such thing. The point is that the statement
``Robert Brown is a wise fool'' isn't true, and it isn't false. It is a
*premise*, an *axiom* from which the argument proceeds. It
happens to be an *inconsistent* axiom, but so what? Who would doubt
that Robert Brown is wise is at least *sometimes* true, or *somewhat* true all of the time. Similarly, who could doubt that Robert
Brown is a fool is at least *sometimes* true, or *somewhat* true
all of the time^{7.7}. Socrates used to do this kind of thing
to poor Thrasymachus in Plato's *Republic* about the same place that
he offered to go shave lions that don't shave themselves^{7.8}.

In any event, if you use a contradiction implicitly, explicitly,
accidentally or on purpose, or somehow insert any *other*
undecideable Gödelian proposition in in place of the contradiction in
a way that performs an equivalent purpose, your final result isn't
``true'', it is a contradiction or it is itself undecideable. In
particular, both ``and'' and ``or'' operations involving a contradiction
are a contradiction. A contradiction contaminates anything it touches,
because asserting it is ``true'' *or* ``false'' gets you into *equal* amounts of trouble, because there are always clever ways of
inserting it into logical statements that corrupt the downstream
processing of the truth tables. It isn't just that the intersection of
your final answer and the set of all true answers is either your final
answer (shown true) or the empty set (from the law of exclusion, shown
false). The law of exclusion is *directly violated* by the presumed
condition, and the *entire result* is null.

Finally, just for fun (in case you, dear reader, have never seen it), consider the following algebra. Suppose I suppose that I have two variables, and that are equal. Then:

(7.1) | |||

(7.2) | |||

(7.3) | |||

(7.4) | |||

(7.5) | |||

(7.6) | |||

(7.7) | |||

(7.8) |

Nothing up my sleeve, right? Or *is* there? Every algebraic
operation is perfectly lovely and valid without question but one, which
has an *undefined exception* that has been delicately inserted into
the proof. The trick line is the one where I cancel from both
sides of the fifth equation. , right? So this is *zero*.
I'm dividing by zero, which is an operation with an undefined answer and
it should come as no surprise that I can prove *one* undefined,
contradictory, null result (that 1 = 2) using another implicitly in the
algebra.

As is the case with computer programs, algebraic or logical or set
operations that lead to NaE or are usually indicative not of a
failure in the ``computational logic'' as embodied in the computer *hardware*, they are not necessarily due to a failure in static *syntactical* validity of the program, they are due to a failure or
mismatch between the program and its *data* - a *domain*
failure. One way to view (and other ) is that they are
*bad data*, ta that cannot exist in the ``Universe'' of set (or
logical, or algebraic, or computational) operations we are considering.

This is an intriguing viewpoint, but (of course) is not original here -
Jaynes points out that many of the difficulties with Aristotelian
reasoning involving the Law of Contradiction and Law of the Excluded
Middle, a.k.a. Western dualism, like the ones that I'm ranting about in
this book come about because one attempts to reason in the incorrect
domain. To be more precise, he asserts that multivalued logical systems
(systems with more than e.g. True *exclusive or* False, *exclusive or* , and so on) can always be reduced to two valued logic
on a larger domain.

I'm not completely certain if I agree. If what he fundamentally means
is that the truth values in e.g. a logical system based on true, false,
unproveable, and contradictory (four possible values all of which appear
like they'd be very useful in analyzing a Gödelian proposition) can be
mapped into a *two* bit binary system (or equivalently that the base
does not matter when you do arithmetic) then sure, I'd never argue with
that. On the other hand, this alone does not make it a ``dualistic''
system as not true does not automatically imply false - it just means
what it says, not (proven to be) true. Also, although there *is*
still a dualistic ``partitioning'' possible - true and not true where
not true includes false, unprovable and contradictory - the *truth
tables* for any given partitioning seem like they'd be very
different^{7.9}.

Jaynes also points out (doubtless correctly, as he was a careful sort of guy) that most of the candidate non-Aristotelian (multiple-valued) systems of reasoning are internally inconsistent, embody logical ``fallacies'' (logicspeak for ``bad logic''), and even argues that predicate logic (the logic of human language involving categorical propositions) is rewritable using the ordinary expressions of algebra and mathematics and is nothing really different from a set theory analyzed with good old boolean/Aristotelian principles in disguise.

However, there are two fairly clear examples where it is not at all
obvious that this is true, and both are well worth mentioning here. The
first is known as *Intuitionism*. Intuitionism basically rejects
the notion that `` is true'' means anything other than `` can be
proven''. For some , the fact that it cannot be proven does *not
mean* that is proven. In this logical/mathematical system, then,
the Law of Excluded Middle is *rejected* as an axiom. Nevertheless,
Intuitionism (as a form of mathematical constructionism) appears to be a
*consistent* form of non-Aristotelian logic where one cannot use
*reductio ad absurdem* - assuming a proposition is *proven*
and showing that it leads to a contradiction, then concluding that
must be *proven*. It is very Gödelian in this respect - a
proposition might well be true (or false) but unprovable.

In real human affairs this is not an empty point - it is, in fact, part
of the basis of the United States Constitution and common law. Just
because I (or ``society'' or ``the district attorney'') cannot prove
that John did not rob the bank does *not* mean that I have proven
that John *did* rob the bank.

Even if it *cannot* be proven by *anybody* - possibly because
there were no witnesses, John left no clues, and John himself is dead,
it doesn't seem to follow that John is guilty because his innocence
cannot be proven. Consequently we require a *direct* proof that
John is guilty, and feel more than a bit uncomfortable with
``circumstantial'' evidence of his guilt or statements like ``John must
have done it because nobody else could have done it''. Does this *really* mean that this particular District Attorney cannot *imagine*
how anyone else might have done it, or cannot prove that anybody else
did it and wants to pick on John? Both are logical fallacies again,
even in Aristotelian logic. Intuitionism carries the avoidance of such
fallacies to the level of the rules of the logic itself so there can be
no mistake - one has to *separately prove or disprove*
independent from respectively, and certain rules of inference are
thereby formally altered.

There is a bit of an empirical cast to this example drawn from human
affairs (instead of mathematics per se), which is fine with me as the
primary theme of this book is *Hume's* (not Aristotle's) empirical
statement of knowledge, that our *sensory stream* is all that we are
knowing of the Universe and consequently Aristotle's actually rather
*absurd pronouncement* as a basis for all knowledge that for a thing
to be known it must be *provable by reason* is - wait for it -
absurd^{7.10}! It cannot even be used as a logical basis
for concluding that we know *nothing*. As we shall see and beat
half to death in the following chapters, although we cannot prove a
single conclusion about the nature of that which we observe in our
sensory stream using reason *alone*, we cannot deny that the sensory
stream itself is *known*.

Jaynes acknowledges this in an indirect way as his entire first chapter
is devoted to the notion of ``plausibility'' - basically an axiomatic
development of the theory of probability to give us *plausible*
grounds for concluding that John *did* rob the bank without the
strict requirement of absolute logical rational proof. This too seems
quite *plausible* to me - it describes the way real humans reason.
However, it doesn't properly seem to appreciate the non-Aristotelian
nature of Hume's empirical statement, and how it is a statement that
describes an *observation* (not an argument) that any of us can make
at any time, the fundamental truth of which cannot be doubted although
it can never be rationally proven. To move beyond this requires axioms,
and axioms cannot be proven rationally either.

Jayne's discussion of plausibility is a very good thing, because this
entire work is devoted to showing that axioms are the basis of all that
we *think* that we know in such a way that it cannot *plausibly*
be doubted (although it cannot be proven as the proposition itself
states). ``Plausibility'' seems to be a better description of human
knowledge. We know *nothing* for *certain* about *anything*
but our instantaneous existential sensory stream. This is true enough,
but useless and (as Hume himself observed) nobody lives that way - we
live instead as if the Universe that we imagine and infer is a *plausible* truth based on some set of axioms that are then more or less
logically developed.

As *The Matrix* movie series so aptly and convincingly demonstrated,
the price we pay for the *logical certainty* of mathematical and
logical reasoning is that it cannot permit any certain conclusion to be
mathematically or logically drawn about Reality. Our sensory stream
could *in principle* not reflect an actual external Reality at all,
or it could reflect an actual external Reality but be mistaken in *every respect* as to its true nature. Indeed a great deal of our
creative energy as a culture is devoted to *making up* or
experiencing Realities that are superimposed ``weakly'' on our minds
through our sensory streams - dreams, interior monologues, fantasies,
hallucinations, movies, music, books, theater, role playing games - all
*mental* experiences that nevertheless communicate to us a *sensory simulation* of a Reality via various means that generate
(externally or internally) sensory impressions ``like'' those that our
presumed Reality itself generates.

*The Matrix* (or its literary predecessor *The Joy Makers* by
James Gunn, or various other works by William Gibson and others) are
science fiction stories that speculate that eventually (due to advances
in technology) it will become possible to make the simulation
sufficiently precise via direct neural stimulation directed by extremely
powerful computers that humans will be unable to tell the difference
between the ``fiction'' being presented to them electronically and
``reality''. At this point the layers of *unknowable abstraction*
between reality as a sensory stream and sets of symbols that *might*
be used as an objective basis for that stream become obvious - the
experience of a thing does not imply its existence. This is not a
purely theoretical argument, as the same thing happens all the time in
our presumably real world. The use of certain psychoactive drugs,
neurophysiological or neurochemical trauma, or plain old psychoses all
can create significant deviations between what one experiences and an
objective reality, sometimes transiently and sometimes permanently.

The second (and probably best known) example of a non-Aristotelian
theory of reason (such as it is) is Alfred Korzybski's *Science and
Sanity* and the umpty derived works by the collective of philosophers
and thinkers who are members of the *Institute of General Semantics*
(http://www.general-semantics.org). This Institute, founded by
Korzybski, promulgates a *semantic* overview of reason that in
certain important respects resembles the viewpoint being advanced in
this work. It is touted as being a *non-Aristotelian* (and
non-Newtonian and non-Euclidean) system of reasoning^{7.11}.

There is a simple mantra for its primary axiom. ``The map is not the territory''. This means that the word for something is not the thing itself. Words are multivalent and categorical, things are unique; therefore ``whatever you say that a thing is, it is not''. Ouch! Seems like this approach has something to say about our efforts to chop up uncountably infinite sets like the real numbers into subsets (where the chopping can be done an infinite number of ways that cannot be specified by any compressed representation - one with less information in it than the points in the set).

However, General Semantics doesn't seem to go this way as information
theory and random numbers aren't their thing. Also, this is pretty
difficult to work through and this isn't a math text. To make this
concrete *and understandable*, let us return to The Universe of
Fruit (as a set theory).

If I wish to sort Fruit into boxes, apples in this one, oranges there,
pineapples over there, I have to pick up a piece of fruit and make a
decision about it. Unfortunately, *every piece of fruit I pick up
is unique!* *This* piece that I pick up is a complex assortment of
objects - sugar, starch, cellulose molecules, pigments in the skin,
antioxidants, toxins, water, alcohol, and more - that are *themselves* complex assortments of objects - protons and neutrons and
electrons, gluons and photons, quarks - in a complex and *dynamic*
relationship that in the not distant past was definitely *not* a
fruit and in the not distant future will definitely *not* be a fruit
but for a brief interval in time has come together into what we call ``a
fruit''. It has other coordinates and properties that contribute to
this categorization - its origin, its size, its genetic encoding -
many of which are examples of ``higher order'' structure yet. *This* piece of fruit is not only unique, every *instant* of its
existence is *independently* unique.

So where do I get off calling it ``an apple'' and tossing it carelessly
into the set-theoretic box at my feet? Even if it was an apple as it
left my hands, quantum mechanics and thermodynamics conspire to more or
less guarantee that it *might not be* an apple by the time that it
lands, that it damn well won't be an apple by any standard at all after
I eat it and excrete it, and that a year earlier my ``apple'' consisted
of a myriad of elementary particle world lines that were gradually being
carried from an initially random state into the *highly organized*
and *extremely transient* state that is different from that of every
other object on the planet that has ever been named apple.

Repeat *ad nauseam* (for every unique object ever given a name,
since by your argument you aren't permitted to make general arguments
because the things they apply to are dynamic and unique) and you too are
now a Master of General Semantics!

Without disagreeing with a word of this (and in fact, duplicating some
of the underlying reasoning here and there throughout this work for an
entirely different purpose) it isn't, really, terribly relevant to
logical systems and systems of knowledge. That is because my knowledge
of the apple, unextended by the axioms of science and inference and
inductive reasoning, unenlightened by language and categorization and
analysis of structure, is appallingly shallow - it is limited to my
*instantaneous* sensory stream in which sensory impressions that may
or may not represent ``the apple'' occur. We are left with a profound
paradox, not of the logical sort but the experiential and rational.

If I see the Universe free of all categories, with every single sensory
impression in my sensory stream being unique and disconnected from any
sort of memory of previous impressions, with no inferred logical
relations between any two parts of the instantaneous stream or the
memory of the stream at different times, then *there is no reason at
all*. The term doesn't apply. Experiencing ``life'' in this manner is
*entirely passive*, and reason leads one to no conclusions
whatsoever. There is no language, as language has no point, no
mathematics because mathematics is not a sensory impression, there is
arguably no *consciousness*, because consciousness itself seems (in
my own mind at least) to involve a complicated feedback process
involving the immediate memory of the past. *Really* treating each
moment as a unique and disconnected experience of sensory data is like
trying to comprehend a movie as a huge pile of its individual frames,
each cut out and flashed up on a screen in a random, disconnected order,
while taking drugs that interfere with the formation of long term
memory. Chaos is the only word for it.

If I on the other hand use language and symbology, if I create *imperfect maps*, if I invent *sets* and methodologies for *sorting out objects into sets and deducing empirical relationships
between the sets*, if I admit time, and memory, and causality, and
physics, and all the other forms of science, then I'm bound to make
mistakes because I casually reason with *imperfectly formed rules*
for specifying sets - I throw an orange in with the apples, literally
or metaphorically, or what I thought were parseley greens turns out to
be aconite^{7.12}. It means that I'm bound to
get all tied up in Gödelian knots when I try to reason about notions
such as ``all men that don't shave themselves'' as this becomes a sort
of a *category error* - the error of putting *anything* into a
category and trying to reason about it, especially a self-referential
one.

Now, I personally am not absolutely certain that General Semantics
qualifies as an actual system of non-Aristotelian reasoning, but it
claims to be one and there *are* well-reasoned analyses of the idea
that in order to sort any sort of real objects (whatever those might be)
*properly* into sets requires an ever growing set of axioms, with a
whole set associated with *each object* that one adds to a given
set. For example, I need an axiom that says something like ``This
particular sensory impression that I'm having, that my presumed memory
and understanding inform me is made up of a myriad of tiny whirling
particles interacting with invisable potentials to create a wave-like
ensemble that represents its collective `state', interacting with a
still more unknown outside Universe that constantly makes small
interactive changes with the ensemble, will at the moment be presumed to
be `an apple' because it has the following inferred properties...''

Only nobody ever says that. They implicitly use broader axioms and live
(or die) with the inconsistencies, if any, that result. So sure, it is
all nicely pointed out. So what. It seems to miss the two main points
- that it *isn't* all about identity of apples, it is about axioms,
because without them we don't even have the ability to talk about not
being able to call an apple an apple just because it is a *unique*
apple and not exactly like any other apple. That ultimately all the
*non-*Aristotelian systems of reason are just as linked to their
axioms every bit as much as Aristotelian ones.

The issue isn't ``dualistic reasoning systems are always bad and
multivalued ones are always good'' as dualistic logical reasoning
systems *work incredibly well* once you've made the right axiomatic
assumptions about the systems you're applying the logic to. It is ``you
have to make a set of axiomatic assumptions as a necessary prior step to
doing any sort of reasoning at all.'' Some of these axiomatic
assumptions will lead to systems that work pretty well as a ``symbolic
representation'' of the time-ordered, memory looped sensory impression
that we call our ``awareness of external Reality''. Others will not.
Some may be Aristotelian and work. Some may be Aristotelian and not
work. Some may be non-Aristotelian and work or not work.

We are attempting the impossible, or asserting the impossible, or
including in our language or symbology the inconceivable, and need to
either restrict the range or domain of certain operations to exclude
metaphorical division by zero or undefined inverse sines or the
possibility of infinite non-convergent loops or reconsider the *meaning* of that which we are trying to compute, or prove, or discuss
altogether. Perhaps the null set is just a symbolic referent that our
Universe isn't well matched with our logical system and ``program''.

However, *algebraically* introducing these very simple operational
*definitions* (not axioms) for a NaE or null set into a naive
existential set theory very naturally eliminates all of the Cantor,
Barber or Russell paradoxes, as the result of the operations proposed or
requested is undefined, or NaE, or restricted away through closure -
the paradoxes do *not exist within the Universal set being
considered*. Precisely as happens in computer science, where the
computer gives you and *error message* that says basically ``This is
a null result, you idiot; fix your code and try again...'' OK, maybe it
leaves off the ``you idiot'' part, but I always imagine it anyway when
debugging my code.

Hopefully by now we have clarified a longstanding puzzle, or word-trap
in the Laws of Thought. The ``state'' of non-being is not a state. The
set of all that are not a set, where is permitted to be any
object in the Universe *including the empty set* is not a set
(even the empty set) in , it is . When the Laws of Thought
refer to ``things'' not being, this is ``the set of all things that are
not in the set algebra'' which is just as self-contradictory as . With this in mind, let us reexamine the Laws of Thought first as
they are written above, then as they are usually *applied* as the
basis for Logic.

The Law of Identity is perfectly understandable as the first grouping of set definitions above. All objects in the set theory live in their own private ``identity set'' which is disjoint from the rest of the objects in the set Universe. The entire set theory is made up of the union of all of these objects. These are the objects that ``have being'', where the empty set is considered to be a set object that ``has being'' within the set Universe.

The Law of Contradiction, which ordinary logic treats as if it involves
the *empty set* (where it would be a vacuous result, as the empty
set cannot contain ``things''), is now seen to refer to the *null
set*. In fact, it is the statement:

(7.9) |

The Law of Contradiction *as stated* is *not* a statement about
applying the ``not'' operation to sets within the theory as if it is
implicitly followed by a set descriptor. It is not (for the set of all
fruit) that something must ``be an apple'' or ``not be an
apple''^{7.13}. It is (for the set
of all fruit) something must be a fruit or not *be* (in the set).
If our entire Universe is fruit, non-fruit is null, not empty, because
*we defined the empty set to be in the set*. We cannot then talk
about cars as if they are ``non-fruit'' so that it is possible for some
``thing'' to be not a fruit - not even the candy-apple red T-bird that
I'll purchase with the vast profits from writing this book if they turn
out to be vast enough - without *implicitly* creating a larger
Universe for our set theory!

In mathematics this sort of thing is pretty obvious. In number theory,
for example, Fermat's Last Theorem (one of my favorite propositions)
states that
has no solutions for integers
for . Of course it is *trivial* to find
solutions for *any* if one relaxes the requirement
that be *integers*. A whole different ballgame results if
we let the numbers be negative (integer or not) and let be a real
number - we are forced to conclude that solutions exist, but they might
be complex. Yet another class of results follows if can be complex.

So when Fermat asserted that this equation has no solutions, that
solutions *do not exist*, he meant that they do not exist *within the Universe of positive integers*. The existence (or
non-existence) of solutions within a *different* set of numbers -
the rational numbers, the real numbers, the complex numbers - is *irrelevant*. They are .

This is a much cleaner formulation. Rather than stating that nothing
can be and not be (leaving the exact meaning of ``something'', ``be''
and ``not be'' ambiguous and embedded in the *dynamics* of being,
with its implicit past and future tenses), we instead create a *static* observation in set theory that the ``set of things that do not
exist within the Universe '' (the null set) has no intersection with
the ``set of things within the Universe '' including the empty set -
the intersection is not just *empty*, it does not exist within the
Universe . The intersection of sets of imaginary unicorns with my
box of apples isn't ``no apples'', it is ``you are out of your mind
considering the imaginary unicorns to be a kind of fruit''. This is a
*stronger statement* (and more accurate statement) of ``non-being''
than the usual Law of Contradiction.

Note that we *ignore* all lesser statements as being *trivial
tautologies* that would never have even been written down in the first
placea if one conjoined to all sets or subsets algebraically at
the beginning, so that the empty set is considered a ``subset'' of all
sets of zero or more members. In that case
leads to *both* that Law of Contradiction *and*
the Law of Excluded Middle as an absolutely trivial application of the
usual definitions of intersection and union, but one is left with no way
to deal with the inconceivable, with the self-contradictory, with .

The Law of Excluded Middle is more troublesome to write as even an
approximately algebraic result. The difficulty is that has *no members* (in the set Universe) and *isn't even an empty set
there*. So talking about ``everything'' being either within the
Universe or not within the Universe only makes sense if there is
a *bigger* Universe in which is embedded, ,
where an object that (such as a candy-apple red T-Bird) *exists* but
isn't in , the Universe of Fruit might be. This is, in fact, the way
the Law of Excluded Middle is usually applied - something must either
be a fruit or not a fruit (in which case it must be something else).
However, Russell paradox sets *cannot* be in such a bigger Universe.
The set of all sets that do not contain themselves in doesn't exist
in some larger , it isn't the empty set (which contains itself
utterly trivially), it is self-contradictory and doesn't exist. It is
*not a set*, in *any* Universe.

Instead let me define the Law of Excluded Middle backwards. Things that
exist (that is to say, everything) are *not* in . Since they
exist, they must be in (by hypothesis, our entire Universe):

(7.10) |

Thus we eliminate imaginary unicorns and red T-birds from the World of
Fruit, integer numbers from the Universe of Pocket Change, complex
numbers from the real line. We also eliminate the Universal
Contradiction from set relationships - the metaphorical or real
division by zero, the , the inverse sine of 2, the set of
all sets that do not include themselves, the set of all Universes where
a male barber shaves all males that do not shave themselves - the *inconceivable* - these exist within *no*
Universes^{7.14}.

Now, how does one get from set theory as the Mother of all Math to *real* mathematics, to physics, to the *good stuff?* At this point we
(as a species, as represented by its brightest scientists and
mathematicians) have a pretty good idea how to proceed at least for
mathematics and physics and the other sciences. I'll devote
considerable space to discussing this in detail in later chapters and
you'll have to just keep reading and trust that I'll get to it
eventually.

The Pit of Existential Despair

Paradoxes

Other Books by rgb: |

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