Pre-Cursor note. Any quotes which are taken directly from Lavine Are enclose in ("") quotes and are taken from chapter IV section I pages 63-76.

Come hear the tale of Axioma, a town outside of time and space. A town which exists on the fringelands bordering the reality field of earth prime. A town of philosophers, angels, devils, mathematicians and other abberations of "normalcy". In recent years Axioma has been forcefully taken over by the tyranical entity Jehovah (also called JHVH-1). This ruler has gripped the land in fear and oppression/ repression. In response to this tyranny a small group has formed an underground resistance. This resistance force is called OMP (Operation Mind Phuck) and is headed by a mysterious figure called the Red Pill. This group has managed to resist the mind numbing power of JHVH-1 under guidance of the Red Pill. We now introduce our main characters of this story:

JHVH-1: The current dictator of Axioma. A tyranical ruler who believes itself to be the supreme being of all of existence.

Conan the barbarian: King of Aquilonia, a land south of Atlantis. Conan is not noted for his knowledge of Mathematics, philosophy or any other intellectual pursuit. Which is why he has attained the rank of general of OMP and held this posistion for 13 years.

Russel: A mathematician/ philosopher, who in this story also happens to be a computer programming expert. He has been chosen to lead the covert operation into JHVH-1's fortress.

Tiles: A mysterious woman who for the most part babbles incoherently to herself.

Cantor: The high priest of ELF (the Erisian Liberation Front), a faction of OMP.

FraterElijah (FE pronounced phi): A neophyte of OMP, who has been noted for his amazing wit, enlightend perspectives and startling good looks.

Hassan: A random factor in Axiomma, for nobody know's whose side Hassan is on. Hassan is also called the Old Man of the Mountain.

Joan d'arc: A spy who has recently been caught by JHVH-1.She is to be burned at the stake for heresy.

Shubu: The legend of Shubu is pure speculation. Mystics and historians have come to know of Shubu by interpreting the ancient texts as existing beyond infinity (whatever that means).

The Red Pill: The enigmatic leader of OMP. During all missions, all operatives of OMP are required to take a special brand of red pills which have reality altering effects.

The GSV (Giant Space Vagina): The legend of the GSV is somehow related to the legend of Shubu. Priests and Priestesses of the GSV have foretold of it's coming, which signifies the end of the world.

Zermelo: A founding magus of Axiomma, who set up laws (also called axioms)upon the reality matrix of Axiomma.

We now start our story in the main headquarters of OMP, where Cantor is just now receiveing a psionic transmission from Joan d'arc.

Cantor: By Shubu! She's been captured!

Conan: Who?

Cantor: Joan, they're going to burn her at the stake on the morrow. Somehow she's managed to break through to me psionically. Here let me patch this in on the main console.

Cantor took some form of helmet from the main console and put it onto his head. This device allowed for direct access to his neural network. The static screen blinked to life with the image of a short haired woman in rags.

Joan: I do not have that much time. I have discovered exactly what JHVH-1 is! JHVH-1 is a renegade satellite created by some alien "intelligence" a long time ago. There was some form of circuitry glitch which caused a logic error in it's CPU when it arrived in Axiomma years ago. This caused this thing to believe that it created all of existence. I have managed to locate the CPU through a hidden entranceway. I also uncovered what the nature of the glitch is, but you guy's are going to have to break in and attempt to reprogram it, and try to save my ass while your at it.

A printout of the location of the secret tunnel was spewed out of the computer with various questions for reprogramming JHVH-1's CPU. The screen then went dead...

Cantor: Oh my god, we've been cut off!

Conan: What do we do?

Cantor: We have no choice. We have the means to correct the glitch in JHVH-1 and the location of the secret entranceway into it's fortress. We must act quickly if we are to save Joan and all of Axioma. Russel, since your the computer expert, you are assighned to the team. Frater Elijah, you must go because of your intelligence op background. Conan you are to provide the brawn in case of trouble and take Tiles along with you for Cannon fodder.

Cantor spoke an incantation and then opened his hand. There were five red pills glowing there.

Cantor: Here. Each of you take one. They'll help a lot.

FE: But there are only four of us going, and you have five pills?

Cantor: Yeah, this last one is for me.

With that each of them issued a prayer to the red pill, popped it, and the party of four left for the fortress of JHVH-1.

Cantor: May Shubu be with you.

The team made their way to the secret entrance to the fortress.

Tiles: Blahh, blahhh, babble gargle, secret entrance.

FE: Yes, Tiles. You really are on top of things today aren't you?

Frater Elijah slapped the blithering wench and immediately shut her up.

FE: Now be quiet!

Conan cleared away the vines covering the entranceway and the party made their way down a long tunnel. There were strange engravings on the walls, showing some form of giant disembodied uterus blocking out the sun, and a strange cat-like figure above an infinity symbol. Tiles was examining the walls very closely.

Russel: We don't have time for this. We must find the CPU. I think it is up a little further.

The party journeyed further, not noticing a shadowy figure taking up the rear. The entranceway to the CPU was ungaurded, as this tunnel was long since forgotten.

Russel: We have to hurry crew. JHVH-1 will be unaware of our reprogramming attempt only for a short while, and I do not know how the answers to the questions Joan gave us will affect it. Conan, you keep a look out, while Elijah and I work on the console. Tiles, well... you just keep doing whatever you are doing (she was looking at the walls). Okay frater, ask me the questions.

Frater Elijah hesitated, shook his head and then said, 'sorry, those damn pills are kickin in. Okay, the first question is located in the theory matrix. There seems to be some missing information on the theory of types. You fill in the information Russel while i'll help you sort it out.

Russel: Found it. Okay here it goes. (Russel was typing away). In the "theory of types, individuals and propositions are taken to be basic, and sentences that apparently mention propositional functions, and classes are analyzed as involving only basic entities." In this sense, the theory is a no-class theory, that is, propositional functions and classes are not taken to be "part of the ultimate furniture of the world".

FE: What the hell does that mean?

Russel: For example, if we have the propositional function, "Joey is human". This can be represented by the pair of the proposistion, "Joey is human" and the individual 'Joey'.

FE: I see, "if we take proposistional functions as basic, we run straight into paradoxes, as is seen by substituing the proposistional function" 'x is self-definable' into itself.

Russel: Yes, but if we adopt my strategy, then 'x is self-definable', must be represented by some pair like, ['Joey is self-definable' & Joey], or [(Joey is human) is self-definable & (Joey is human)].

FE: Shouldn't this also depend whether we are taking the variable in 'x is self-definable' to range over individuals or propositions.

Russel: Hmmm, neither of these captures our original intent, as the variable was supposed to range over propositional functions. We'll have to make the one variable become two. Like '(x,y) is self-definable', where self-definable means something concerning the pair of variables (x,y).

FE: And (x,y) is self-definable is a propositional function that is to be represented by a triple. [remember there are 2 free variables here].

Russel: So in the above case of Joey, we would have: (Joey is human, Joey) is self- definable; (Joey is human), Joey. And we do not even have to make sense out of 'self-definable' because self-definable was defined to be for propositional functions represented by pairs.

FE: Yeah! And the propositional function we intended to plug in and get paradox is represented by a third.

Tiles: So the paradox is not allowed because we cannot admit the paradox by definition.

Russel and Frater Elijah looked at each other dumbfoundedly. How did Tiles know? She just pointed at the wall where there were strange markings:

Russel: There may be more going on here than meets the eye, frater. Let's go over what we have so far.

FE: Well, "the analysis of propositional functions into propositions and individuals creates in effect a hierarchy of types, with propositions & individuals at the bottom, with single variables ranging over them, then propositional functions of propositional functions or individuals with pairs of variables ranging over them," ad nauseum...

Russel: The effect is that no propositional function can exist in it's own range, and we can avoid a paradox loop. It looks logically pure to me.

FE: To block the paradox, you've introduced "orders" of propositions. First order propositions at the bottom, then second order and so forth, each order level can involve quantification over those of some order. "Quantification is not allowed over all propositons, but only over propositions of a given order."

Russel: Seems adequate to, i'll label this file "ramified hierarchy".

FE: There is still somthing that does not feel right.

Tiles started convulsing in the corner. Her eyes rolled to the back of her head and she spoke out in the voice of an elderly man (obviously she was possessed). It was the voice of Zermelo himself. He spoke in a garbled tounge and then pointed to a section of the wall with strange markings on it:

Russel: Of course, the Axiom of Reducibility! It seems that a finite class is one such that there is no one to one mapping between a strictly proper subclass, and we cannot use this fact to define finite classes because we can only quantify over some order. We can't express, "no functions what-so-ever" only "no functions of such and such an order", that is, "we are faced with the possibility of a class being finite with respect to functions of some order, but infinite with respect to functions of a higher order." So we cannot do ordinary mathematics.

FE: It appears that what is missing is a notion of class which does not have an associated order.

Russel: The paradox which leads us to introduce orders was intensional, while classes are extensional. By the addition of the Axiom of Reducibility, we "can assume that some order includes so many propositional functions, that they can represent all classes. Since all of the classess will be of the same order, we shall be able to quantify over all of them." Thus are we able to develop mathematics in the theory of types.

Tiles: (in the voice of Zermelo) ~ Yes, Russel you are wise, but there are still gaps. For one, it is impossible to prove the axiom of infinity (that a class has infinitely many members) and two, we cannot prove the axiom of choice. But even these are less serious than what is imposed by the Axiom of Reducibility. The Axiom of Reducibility would have to be so universally accepted and used that "even natural number theory (let alone the rest of mathematics) would depend upon it as a hypothesis, since it is required to give an adequate definition of finiteness."

At the above Frater E. was completely baffled. For one thing where the hell did Zermelo pull the other axioms out of. Our beloved frater deemed to remain silent though in this interlude.

Tiles: So before you push the [enter] button, you may want to consider your modification to JHVH-1's CPU. JHVH-1 does control all of Axiomma and you know not what may happen.

Russel: He's right. But before I do anything tell me of your position Elijah so I may weigh my options.

FE: A lot of this discussion Russel is out of my hands due to my particular take on mathematics. It seems to me that a logical basis for mathematics is highly unlikely, due to the nature of existence.

Russel: What do you mean?

FE: We've been talking back and forth with working tools called propositional functions and logic. We have even been blessed with a visit from Zermelo himself (possessing Tiles). In giving an account for mathematics however, we must first assume that we are talking logically. Where 3 does not equal chicken, but 3 = 3 and this is supposed to make some kind of sense. I believe that an account for mathematics is an account for the rationality of existence (in extension of course). Which we can work with on an accepted level but this only possesses any meaning insofar as we say it does. In this context what your theory offers us is increased tediousness by the addition of the reducibility axiom, but this is as good as anything else which works. It does not particularly mean anything in and of itself. So whether we accept it or not, to account for mathematics, depends totally on ones ontological stance to begin with. No one being more correct than another (except maybe in usefullness). For Nothing is True and....

A black clad figure stepped out of the shadows at that very moment, revealing himself to be...

Hassan: Everything is Permitted (finishing Frater Elijah's sentence).

With that one sentence said, all answers shall follow.

An alarm went off, gaurds appeared out of nowhere. Conan was swearing to CROM, Elijah stood face to face with the Old Man of the Mountain, Tiles Reverted to her incoherant state, and Russel was at the terminal with his finger over the enter key.

Russel: There is no time! JHVH-1 knows were here. I don't know if my theory will repair the glitch or not. Oh, Joan can you ever forgive me???