Come hear the tale of Axioma, a town existing on the fringelands of earth primes reality field. A town of angels, demons, mathematicians, and other aberrations of normalcy. Before beginning this final tale, let me introduce our cast of characters.

• Frater Elijah (FE pronounced phi) – A neophyte of OMP, who has undergone many adventures into the unknown spaces. Our good frater is the hero of this last tale.

• WE – An alternate personality of our good frater, which takes over now and then. WE causes all manner of havoc and ontological anarchy whenever they manifest.

• Aleister (Alexander) Crowley (666) – A renowned magus, pervert and drug using philosopher. He is a beastly individual accustomed to all manner of indulgent vulgarity. In addition to being one of the greatest magi of this century, he was also a superb writer and poet.

• Ouspensky P.D. (PD) – A mathemagician with mystical tendencies who engages in all manner of weirdness un-becoming of the normal mind.

• Lavine (L)– A mathematical philosopher who is head of the fundie cultus of Axioma, which is bent on understanding the infinite… rationally.

• Shubu – The beanie baby which exists beyond infinity. Shubu is akin to Tao (whatever this may be).

• The Giant Space Vagina (GSV) – A disembodied Uterus which floats through space. The legend has been explained in previous essays.

• JHVH-1 – A dysfunctional megalomaniac satellite which was once ruler of Axioma.

• Bassle (bass)– A brilliant man capable of great artistic appreciation as well as possessing a keen intellect, which is only balanced by his vast depth of knowledge. Bassle has the power, kindness, and wisdom to pass all of his students with high grades for their exceptional astuteness.

• Seany (S) – A typical avant-garde philosophy student, who under an urge to be different complex, has decided to take a philosophy of mathematics class. The reincarnation of Ouspensky on Earth Prime.

• Pat (P) – Another typical philosophy student who feels himself to be different from the rest of humanity. The Reincarnation of Lavine on Earth Prime.

• Karina (K ~ This does not stand for ketamine)– A girl with a ferret.

• Alex – The reincarnation of Frater Elijah on Earth Prime. Our good frater is commonly mistook to be a Gnostic.

Axioma has been completely and utterly destroyed. JHVH-1 blew a gasket when it realized that is did not own a monopoly on the universe, and with the ensuing chaos (not to mention the giant vagina descending from outer space); Axioma was completely reduced to intellectual rubble. We now begin our story one hundred and eleven years after the return of the GSV upon Axioma and in a small classroom on the second floor of a library on earth prime. A class is in the middle of a heated debate.

Bass – Yes, and as I was saying. I believe it is here that Lavine's argument is breaking down… (glancing up at the clock anxiously)… but we’ll have to save that for next time, as we are out of time.

The class started to dissemble and Sean, Patrick, Alex and Karen assembled for one of their brief after class chats.

S – Just when i’m beginning to think that I have a grasp on this stuff, Bassle always pulls the rug out from under me.

Alex – Yeah! He sure is a genius, right!

K (while playing with her ferret) – Yeah.

P – I find all of this most disturbing.

Alex – What do you mean?

P – It seems that we have not come to a conclusive finality on the continuum hypothesis.

This highly upsets my understanding of the universe.

Alex – Ha! As if we understand anything about the universe!

K – You guys…

S – You know. Something really interesting happened the other day. I was cow tipping with some old friends of mine from southern Georgia and we came upon this patch of red fungus. It was strange, because I never saw fungus like this before. It sort of looked like giant pills. I figured that maybe we could go back to my place, eat this fungus up and talk some philosophy. Do you know what I’m saying?

Smiles spread widely across all faces at the mention of such an offer, free of charge…

Sometime later at Sean’s residence, the party consumed the bountiful sacrament and things started to slip away…

P – Okay let’s get started with this discussion.

What the group did not know was that Alex was way beyond the point of coherency at this point and things started to morph… explosion!

It all came back in a tidal wave of memories.

Fade now to a memory sequence…

It was just after the GSV has destroyed Axioma. It was an awful quivering thing, best not to look at. Our good frater was standing before something that was beyond his grasp. This terrible, terrible uterus quivered in an almost incomprehensible language.

GSV – I will hear you case now. You are to present to me a case concerning the infinite. Since your kind have sought comprehension of that which I serve, it has heard your cries in the dark and now wishes to hear what you have learned. As being a judge of the glory transcendent, I am merciful and shall allow only a handful of you access through me. You can call upon three other individuals to offer a diversity of views. If any of you are worthy you shall ascend, if not, you shall be re-newed.

Being the wise frater that he was, Elijah decided not to waste anymore time and get on with the discussion. Our handsome hero opted to go last in this discussion.

FE – I call upon Lavine, the Beast 666, and Ouspensky. In addition, I shall engage in this presentation.

The three individuals appeared, immediately cognizant of what was transpiring. Lavine went first.

L – Previously, I have attempted to show that "we know about the combinatorial infinite by extrapolating from our experience of the indefinitely large" (Lavine, pg. 288) (see also essay 2). I firmly believe that to this end, "the principles of finite set theory can be motivated as self evident principles concerning finite sets, including indefinitely large ones" (pg. 288). Without this, we could view the finite as extrapolated from the infinite, in stead of, the infinite extrapolated from the finite.

FE – Much the same way as modern Newtonian physical laws can be interpreted both ways, giving rise to rational paradox. The key word here is rational.

L – Precisely, "what has happened is that we have systemized our experience of the indefinitely large without attending to the context-dependent features of that experience, and this has led to our picture of the combinatorial infinite" (pg. 288). With our notion of finite mathematics (essay 2) firmly in mind, we shall talk of Fin(ZFC) qualitatively, in light of Cantor’s approach to the transfinite. Now Cantor seems to have had the tendency to attribute characteristics of finite collections to infinite collections.

FE – Yes, these are called transfinite collections. Somewhere between the finite and yet not quite infinite.

L – Now the distinction between the transfinite and the infinite (absolute) is similar to the difference between sets and proper classes. This is also similar to the difference between finite (including indefinitely large) and infinite. "Let me remind you that the classes we are considering are combinatorial classes – collections that are too big to be counted (counted within the present universe, we might now add) – not Russellian extensions with their attendant paradoxes" (Lavine, pg. 319) (see also essay 1). The Ordinals comprise a Proper class.

FE – That sounds like Cantors reasoning, you rip off.

L – I am just claiming that cantor was working out some form of transfinite progression, such as 0, 1, …, ¥ , ¥ +1, ¥ +2, …, ¥ *2, …, ¥ *3, …, ¥ ², …, ¥ ³, …, ¥ ^ ¥ , …

PD – I see a problem already but I shall wait for my turn.

L – It seems that Cantor thought to make sense of infinity, by bringing in infinite progressions represented by indefinitely long ones. On a side but related note, it is obvious that very attempt of motivation in this area is for potential manipulative capabilities; the idealized (Cantorian) notion of a an Omnipotent Mathematician.

At the mention of potential manipulation of infinity, the GSV made an awful squooshing sound that was not for sober ears.

666 – Be careful Lavine, you tread on dangerous ground here." Proof is only possible in mathematics, and mathematics is only a matter of arbitrary conventions" (*3).

Lavine shrugged off the comment with typical arrogance and a cutting smile.

L – Where was I? Oh, back to my extrapolation, "it is undoubtedly the case that combinatorial sets share many features with finite ones. But to see how they come to share those features, we need to look at extrapolation from the indefinitely large," not some Omnipotent Mathematician (Lavine, pg. 290). Mystical Crap.

FE – Can you explain once again the difference of indefinitely large vs. infinite vs. transfinite in this context?

L – Infinity, I shall not touch yet, because that is what we are trying to "get to". The transfinite does not corresponds to my notion of finite mathematics, but the indefinitely large is the finitist equivalent (in not so many words) of the transfinite. The transfinite as stated previously exists after the natural numbers, but they are not quite infinite. Now, "the idealization of those capacities (*4) to infinitary ones is part and parcel of the extrapolation from the indefinitely large to the infinite" (Lavine, pg. 290). Now, concerning a specific discussion of Fin (ZFC), all of the axioms prove true in the theory itself, that is, they are true of finite sets, with the exception of one Axiom.

FE – Let me guess [sarcastically], the Axiom of Infinity?

Frater Elijah looked on with disdain as Lavine attempted to reduce the notions of infinity with simple twists of mundane logic and language.

L – Correct again. For the Axiom of Infinity in finite set theory, there is the equivalent called the Axiom of a Zillion (AOZ). The AOZ says that there are indefinitely large sets. "Theories in finite mathematics all use indefinitely large domains to bound the quantifiers" and "the indefinitely large W ’s of finite mathematics replace the fixed infinite domains of ordinary mathematics, and they permit us to have many of the advantages of quantifiers that range over an infinite domain" (Lavine, pg.291). So we have the AOZ asserting the existence of a indefinitely large collection inside of an indefinitely large domain, without the problems of infinity. We have experience of indefinitely large collections (*5)and so this is all tied to mundane experience. So I have shown (in a rough manner) that finite set theory can be motivated in this way. I would attempt to check all of the axioms of Fin (ZFC) reflecting this motivation but this is quite extensive and fear it beyond the scope of my audience. Not due to lack of intellect, it is just that "the axioms of the final precise system are transcripted into an awkward formal language of something preformal expressed in natural language" (Lavine, pg. 292). This is all symbolic representation.

GSV – I wish to point out Lavine that Cantor’s initial supposition was that "the Absolute can only be acknowledged and admitted, never known, not even approximately" (Lavine, pg. 52). But enough from you Lavine, Ouspensky since you are somewhat of a mathematician-mystic, you go before Crowley and our good frater, whom I feel may go off the deep end.

PD – Infinity is "the only reality and at the same time it is the abyss, the bottomless pit into which the mind falls". This is inevitable as a mind approaches a higher reality. As the mind wakes from it’s sleep the vision becomes clearer and clearer only to realize that one is lying down in the dark. I speak no in loose and abstract words here. Let me start at the beginning. Let us consider an animal, who is starting to get brief flashes of human understanding. Formally, the animal thought is terms of:

There was separation of one thing from another, they could not distinguish that all houses are houses. If one tries to prove that the animals house and the strange house (from the above case) were both similar, that they are both houses, the animal is confused and adamantly refuses this relation. "The idea, this and that are similar objects, the articulate two-dimensional being will translate into the language of it’s logic, in the shape of the formula: this is that; and of course will pronounce it an absurdity, and that sensation of the new order of things leads to logical absurdities" Thus it will be unable to express what it sees with it’s own logic. This is what happens with the case of infinity. We need a new logic to undergo our explorations (Ouspensky: T.O., pg. 219-222). 

L – I don’t understand.

666 – "The poorest tongues have wealth of Concrete’s"(*10).

FE – Of course you don’t. I believe this is similar to Russell’s order of types, except that it relates to thinking & possibly consciousness.

GSV – Relate this specifically to our notions of the transfinite please.

PD – In due time my dear er…organ. Mathematics can be viewed as the science of magnitudes, whereas logic can be viewed as the science of concepts (*7). These are not definitions, they are only translations. Both logic and mathematics built up out of our notions of our interaction with the phenomenal world. These interactions give rise to laws. The logic (Aristotle, Bacon) of this formulation deals with concepts only. "Logos, the word, is the object of logic. An idea in order to be substantiated to the laws of logic, must be expressed in a word. That which cannot be expressed in a word cannot enter into a logical system". Also a word which enters into a logical system can be subjected to it’s laws only in the context as a concept. We all know that there are things which cannot be expressed into words and therefore do not enter into the realms of logic. This shall all lead to our uncovering that "mathematics, the most exact of sciences, is entirely illogical" (Ouspensky: T.O., pg. 222-223).

L – That is a bold claim.

PD – Not really, comparing the axioms of Aristotelian (in addition to Bacon) with the Axioms of Mathematics (common interpretation) we find a great degree of similarity:

"The Axioms of Logic Mathematical Axioms of Difference & Identity

are themselves equal"

Both the laws of logic, and the laws of mathematics extend from our interaction with the world around us and thus our reasoning faculties. "Just as the axioms of logic deal with concepts only, and are related solely to them, so the axioms of mathematics apply to finite and constant magnitudes only, and are related solely to them" (Ouspensky: T.O., pg.223).

FE – Are you implying that the infinite is governed by different laws of mathematics.

PD – Not only that, but by a different logic of understanding which appears irrational to us. I feel that our understanding of finite mathematics and the laws governing them arrising from our interaction with the world is a fairly easy concept to grasp so now let me explain the mysteries concerning infinite magnitudes and variables. We shall enter the realm of the transfinite (*8) with the real numbers. Infinity which is represented by the ellipse is an expression in mathematics with which we can perform all mathematical operations (we do not need to make explicit sense out of this yet). Now it is possible to raise infinity to a power such as ¥ ¥ . This magnitude is greater than the simple ¥ and yet at the same time they are both equal ¥ ¥ = ¥ . "This violates the fundamental laws of mathematics accepted for finite numbers. After a change (*9) the finite number cannot equal itself," but as you can see that after a non-trivial transformation the transfinite number remains equal to itself.

FE – It’s like they absorb quantities. Like for example 5 times infinity is still infinity.

PD – Precisely good frater, any distinction at this point does not do us any good.

L – What about reality?

PD – I doubt you have any conception of what reality really is but I think I understand what you mean. "All transfinite numbers are entirely real," and we can find examples of them in our world today. We know that there are an infinite number of points in a line. Now take a line 1 inch long & a line 1 mile long. Both lines have an infinite number of points, and we again get ¥ = ¥ . Let’s look at a solid square now. One side of the square is composed of a given line segment, and there are an infinite number of lines in the square and the number of points in each of these lines in infinite. So the number of points in the square is equal to ¥ ¥ . This is obvious greater that ¥ and yet at the same time they are equal.

FE – Your logic seems similar to Cantors Cardinal Numbers.

PD – I must assert that "all infinite magnitudes are equal, because if there be an infinity, then it is one, and cannot change". We can blatantly see that the fundamental axioms of mathematics do not hold for the transfinite. Now here is a key point. "We may also say that the fundamental axioms of our mathematics are valid for constant magnitudes only". They demand unity of time and moment and are equal to themselves only at a given moment. If we look at a magnitude which is changing then at another moment it cannot equal itself. "But this is precisely the thing that I am talking about. The axioms of our mathematics are applicable to finite and constant magnitudes only. Thus I assert that since everything is in a constant state of flux, the mathematics of the finite is unreal. I can say now that the fundamental axioms of mathematics are true now only under artificial conditions which do not exist in nature. In nature there are no concepts and there are no finite constant magnitudes. (Ouspensky: T.O. pg. 225-228).

L – But if infinity is unchanging as you say? And if this infinite mathematics models the real reality as you are implying? This is a paradox because reality is changing all the time.

PD – There is no paradox here. You are still thinking like our two dimensional animal. The infinite is the totality of all. There is no changing of itself. We can only visualize and see sections of infinity and thus we see change. This is a result of an imperfection in our space-sense

GSV – Enough! I have heard enough from you Ouspensky. Crowley, you now present me your view.

666 – I shall do more than present my view of the infinite, I shall present the entirety of existence.

GSV – Intriguing. Proceed.

666 – In the beginning we have the negative, the tripartite zero as absolute negative existence. Nothing is distilled from this 000 to form a basis of Nothing, the dualistic 00, and it is here that we have the birth of chaos as the pregnant void, 0. All of these are Nothing in Particular. In this Nothing, an idea a flash. The Unity, The Self of Deity beyond Father and Mother, a ray of light erupted the darkness. One reflected itself which alludes a dyad 2. Now "3 reconciles 2 and 1, here are placed the aspirations of divinity". The 3 is obtained by the addition of the 1 & 2. A reflection of the triad spawned by a it’s predicament across the abyss. Now separate it cannot achieve unity of itself with Itself in Unity. There is no other direction to go except more fractalization. (Crowley, #)

FE – A brilliant synopsis of the upper half of the tree, oh beastly one, but I fear that the uninitiated are duly lost now.

L – What?

666 – "We have seen all numbers as Veils of the One, emanations of and therefore corruption’s of the One. It is the Universe as we know it, the static Universe" (Crowley, #). The magus finds no satisfaction in contemplating this arrangement, for they are all divorced from the one.

FE – I believe I can see an analogy here between the possible construction of all numbers from zeros and therefore 1’s. But this oversimplification is only for the academics.

L – I have no idea what the hell he is talking about. If when he says the One he means god then, I would have to protest that I am an atheist.

666 – I don’t know why I even waste my breath. Trivial issues for trivial idiots. Dearest frater let me warn you on your journey of these atheists of which there are three kinds:

1. "The mere stupid man. (Often he is very clever, as Bolingbroke, Bradlaugh, and Foote were clever.) He has found out one of the minor arcana, and hugs it, and despises those who see more than himself, or who regard things from a different standpoint. Hence he is usually a bigot, intolerant even of tolerance".
2. "The despairing wretch, who, having sought God everywhere, and failed to find Him, thinks everyone else is as blind as he is, and that therefore if he has failed – he, the seeker after truth! – it is because there is no goal. In his cry there is pain, as with the stupid kind of atheist there is a smugness and self-satisfaction. Both are diseased egos."
3. "The philosophical adept, who, knowing God, says "There is No God," meaning "God is Zero," as qabalistically He is. He holds atheism as a philosophical speculation as good as any other, and perhaps less likely to mislead mankind and do other practical damage than any other" (Crowley #).

I would like to say one more thing before closing.

"IT moves from motion into rest, and rests from rest into motion.

These IT does always, for time is not.

So that IT does neither of these things.

IT does THAT one thing which we must express by two things neither of which possesses any rational meaning.

Yet ITS doing, which is no-doing, is simple and yet complex, is neither free nor necessary.

For all these ideas express Relation; and IT, comprehending all Relation in ITS simplicity, is out of all Relation even with ITSELF.

All this is true and false; and it is true and false to say that it is true and false.

Strain forth thine Intelligence, O man, O worthy one, O chosen of IT, to apprehend the discourse of THE MASTER; for thus thy reason shall at last break down, as the fetter is struck from a slave's throat" (*11).

GSV – Oh great beast, I marvel at thy wisdom, there is no more need for you to continue.

L – But?

GSV – Quiet! It is now our beloved frater’s turn to share his views of the infinite.

FE – I have heard arguments from a purely mathematical standpoint, a purely mystical standpoint, from a middle ground between mathematics and mysticism (Lavine, Crowley and Ouspensky respectively). What I have to offer in this was is something entirely different…

At this time our beloved frater started to convulse and his countenance became as a madman who has been in school too long, and written way too much in the appeasement of others. Our beloved Frater has transformed into WE. A schizophrene of the utmost wisdom. What the GSV and nobody else knew, was that WE has stolen the feather of MAAT (*12). The Beast and Ouspensky had the wisdom to jump into the GSV while they had the chance and vanished into it’s many folds. Lavine was to caught up in trying to rationalize everything and just stood dumbfounded.

WE – You see dear Lavine and my beloved Vagina. Truth cares not for words. There is not a problem with the Continuum at all. All numbers are infinite. By my wit and my might have I stolen this feather of Truth, and with it all shall be undone. The transfinite is of good a use as the finite for there is no difference. The only problem lies with your lies. The infinite is of yourself and you are of the infinite.

With a wave of his hand and the feather of Truth WE sent a wave of destruction across all of Axioma, destroying Lavine (again) and banishing (*13) the GSV to outer space again. Realizing their predicament now and the futility, he opted to incarnate over and over again across many spheres and dimensions. With this Shubu Smiled.

When Alex came to, he still saw his class mates still in a discussion concerning omega’s an progressions and the transfinite. But now, Alex was in full possession of what he was and where he (or shall I say WE) has come from. With this he promptly got up from the room, left without a word, and set himself as an angelic hero to be the enemy and witness to Truth.

What Alex did not hear, was the chanting which emanated from Seany’s house later that evening.

• Crowley, Aleister – "An essay upon number" (Sure Fire Press, 1988) ISBN# 1-55818-106-7. Original date of Writing (?), it is a 14 pg. essay available through HOLMES Publishing group. P.O. Box 623 Edmonds, WA 98020. Abbreviated #.

• Crowley Aleister – "The Book of Lies" (Samuel Weiser). This text is readily available. Check out http://netropic.speakeasy.org/crowley/

• Lavine, Shaughan – "Understanding the infinite" (Harvard University Press, 1994) ISBN# 0-674-92117-8. Concentration on pages 288-293.

• Ouspensky, P.D. – "The Psychology of Man’s Possible Evolution" (Vintage Books, Jan, 1974) ISBN 0-394-71943 Various idea’s.

• Ouspensky, P.D. – "The Symbolism of the Tarot – Philosophy of Occultism in Pictures and Numbers" (Dover Publications Inc 1976) Library of congress number 75-31286 ISBN 0-486-23291-3 Various idea’s.

• Ouspensky, P.D. – "Tertium Organum ~ A key to the enigmas of the world" (Vintage Books Edition, Sept, 1970) Library of Congress Card Number: 76-128440. Concentration is Chapter 20 concerning infinity and transfinite #’s. Abbreviated T.O.

(*1) verse 14 of "The Book of Lies".

(*2) - The Chaldean Oracles of Zoroaster.

(*3) verse 45 excerpt from "The Book of Lies".

(*4) – Capacities here refers to the manipulative capabilities for indefinitely large sets. This is possible because the indefinitely large is finite. There seems to be a possible smuggling in of the infinite here under a manipulation of language. This reflects a great deal the historical epistemological notions in the formulations of mathematics at a whole, by the manipulations of language (for both boon & bane). Please read my masters thesis for a more enhanced view of this.

(*5) – Such as the number of sand grains on the beach (see essay 2).

(*6) – See again my masters thesis.

(*7) – This is our intuitive notion of concept. Of special note is according to Ouspensky a concept is a logical unit which possesses all the properties of a finite & constant magnitude.

(*8) – Ouspensky defines the transfinite as "numbers beyond infinity" (pg. 224 T.O.).

(*9) – This is of course meaning, not the identity mapping.

(*10) – Verse 24 exert from "The Book of Lies".