Come hear the tale of Axioma, a town outside of time and space. A town that exists on the fringe-lands bordering the reality field of earth prime. A town of philosophers, angels, devils, mathematicians and other aberrations of "normalcy".
Following is a brief review of our beloved cast of characters:
New to our story we have the following characters:
When last we left off the fortress of the tyrannical JHVH-1 was infiltrated by a small band of brave warriors who were attempting to save all of Axioma by reprogramming JHVH-1 and possibly save Joan d'arc in the process. The preceding story finishes, and this one starts with the ending and beginning melded together for coherence (was that coherent or what).
FE (Frater Elijah): 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 on ones (desired) ontological stance to begin with. No one, being more correct than another (except maybe in usefulness). 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 this one sentence, all answers shall follow.
An alarm went off; guards appeared out of nowhere. Conan was swearing to CROM. Elijah stood face-to-face with the old man of the mountain while Tiles reverted to her babbling state of incoherency, and Russell was at the terminal with his finger over the enter key.
Russell: There is no time! JHVH-1 knows were here. I don't know if my theory (of types) will repair the glitch or not. Oh, Joan can you forgive me???
With that, he pushed the button… Silence … Darkness… the only thing that was visible was a glowing symbol on the chamber wall:
¥
The symbol mutated and warped in a strange black light. It formed into the following symbol:
T
Streaks of lightening erupted from this enigmatic symbol. Flashes and screams echoed throughout the chamber. The scene was chaos (hehe). There was no telling how much time passed, but Frater Elijah and Hassan were the only ones that were spared the massacre that followed. Russell was apparently fried at the terminal, the only thing left was a charred husk, with the word "type-cast" emblazoned on his forehead. Conan was nowhere to be seen. Zermelo's ghost apparently could not keep hold of Tiles meandering consciousness and departed. Thankfully Tiles was dead in the corner and will trouble our story no more. All of JHVH-1's guard's destroyed by the chaos. A hole in space-time opened up and two figures stepped out.
Hassan: Good to see you again Lavine, I see you brought one of your lackeys with you.
FE: What happened.
RP (Rishi P.): It is a strange coincidence that we were attempting to manipulate JHVH-1's memory at the same time that Russell and you were adding your re-programming. The combination of the two has caused some form of backlash, hence this mayhem.
Hassan: Where are we?
FE: I haven't the foggiest.
Lavine: We were skirting the CPU of JHVH-1 using some new technology which allows us to reduce ourselves to the size of infinitesimals. But something went wrong when both our programming and Russell's collided. We barely managed to get out of the CPU alive. The trajectory took us to this place. Rishi P. You have the reality map. Where are we?
RP: We are nowhere?
FE: By Shubu! All of this discussing and we have gotten to Nowhere.
The characters looked around and noticed sections of nothing eating up existence.
A hissing voice echoed in all of our characters heads.
Nobody: Thiss iss corrects. Yous ares nowheres ands nothings wills haves yous.
FE: I don't know about you guys but this place is scaring me.
Hassan: 'Tis no bother really. I have been nowhere before. But I fear for their ilk (he motioned to Lavine and Rishi P.) it spells disaster. You see fundies cannot live inside of nothing and they dissolve.
FE: I don't like the fundies (a.k.a. fundamentalists), but I don't think we should just let them die. I mean they are fun to laugh at sometimes, right?
Hassan: You know your right Elijah, they are fun to laugh at but saving them sounds less fun than watching them face Nothing.
With that Hassan faded into the nothing of the surroundings.
FE: Okay Lavine here it is. Maybe if you start telling us something, then we can get you guys out of here. It should also form some sort of bridge between the infinite and the finite (because that is what got us nowhere to begin with). I have an extra stash of red pills, since I know both of you are fundies and don't take pills, I'll take all of them. Maybe it'll help me by some of what you guys are talking about. Okay, shoot.
Lavine: Okay, I'll try and give a very brief summary of my claim that we can understand the mathematical infinite. In doing this I will try and convey what role finite mathematics plays and some of my fundamental assumptions. Rishi P shall offer his expertise in physics and I am sure you Elijah shall offer your two cents. I start by saying that "we cannot deduce or falsify a proposed new axiom that settles the Continuum hypothesis on the basis of known axioms… we therefore need a non-deductive, non-axiomatic way of determining whether a proposed new axiom follows from our old standards" (Lavine, pg. 244).
FE: This is all for understanding the tru..(Elijah almost said that damnable word) er.. the validity of the Continuum Hypothesis.
Lavine: Yes, "we want to know whether or not there is a genuine subset of the genuine real numbers that cannot be placed into any one-to-one correspondence with either the genuine natural numbers or the genuine real numbers" (pg. 243).
RP: Master, maybe you should not use the word genuine so much. You sound like you don't believe yourself.
Lavine: Hush up. Okay, "we have already seen that the infinite is nowhere to be found in reality" (pg.246)
FE: I ain't gonna touch that one.
Lavine: It seems that Cantor's developmental progression to the transfinite is highly appealing. {Lavine was scribbling on the floor now}.
0, 1, …, oo, oo+1, oo+2, …, oo*2, oo*3, …,oo^2, …, oo^3, …,oo^oo, etc… Why does this progression make any sense to us at all? We even represent infinity by these ellipses.
RP: "Since modern physics makes it seem likely that the physical universe is of finite extent and decomposes into parts with a non-zero minimum size" (pg. 247)
Lavine: Yes, so where exactly does our notion of the infinite (and that represented by ellipses) come from?
FE: Is that like the question of where do babies come from?
Lavine: Your wise ass remark serves my purpose. Yes, it is similar. Much similar to the way children grasp notions of infinity as being, "too large to count". We first extend our notions of large to indefinitely large, so "my proposal is that it began with the extrapolation from experience of indefinitely large size" (pg. 248). Our idea of indefinitely large comes from our experience of things that are to big to count, like the grains of sand on the beach.
FE: Or the number of morons in the world.
Lavine: Exactly! Now with my child analogy, "the child is not wrong that an infinite quantity is one that is too large to count. But the child might mean that it is too large to count today, or so large that even Daddy won't be bothered-too large in a context-relative sense" (pg. 248).
FE: What are you proposing exactly?
Lavine: I am proposing that the "infinite is nothing more than too large to count-too large in a context independent sense, too large for anyone to count independent of context, abilities, or interest" (pg.248).
Nobody smiled and noticed that the nothing was spreading faster towards the fundies as they discussed…
RP: How shall we extrapolate this in a mathematical sense?
Lavine: Good point. We extrapolate by dropping the context dependent states, similar to dropping the subscripts of indexes for a sequence of omega's (as in the ordinals --> cardinals). Extrapolation from the theory of indefinitely large does provide a rational justification for believing our theory of sets. To sustain my claim, three things must be shown. "The first is that the informal theory I present as a theory of indefinitely large size is in fact nothing more than a precise version of some of our intuitions that arise from experience of the indefinitely large [*!]… The second thing I must show is that the twin notions of indefinitely large size and extrapolation are sufficiently rich to bear the heavy explanatory burden I am placing on them [*!@] …The third and final thing I must show is that the notion of infinite size that results from extrapolating from the notion of indefinitely large is the one that actually was operative in the development of modern axiomatic set theory [*!@#]" (pg. 249-251).
FE: I see a problem already with your notion of intuitive understanding, but for the sake of argument I remain silent.
Lavine: For the showing of [*!] we assemble a theory of the indefinitely large using the following properties to sidestep the use of the infinite. "Only finite entities are employed, and all quantifiers have finite ranges" (pg. 252). This entire theory can be described without using any quantifiers in the infinite range. "Nothing infinite, whether actual or potential, is either used or presupposed by the theory" (pg.252).
Nobody: Nothing infinite indeed. Hahahahaha
Lavine: As I was saying, this theory is free of infinitary presuppositions. Very similar to Hilbert and the work of Mycielski. These theories that are free of suppositions of the infinite I shall call finite mathematics.
FE: Please be as simple as you can and leave out the specifics. Only the generalities concern me here.
RP: "Finite mathematics is just more mathematics, and it has no more need to confront ontological questions than any other mathematics" (pg. 254).
Lavine: Rightly so. Now I skip a direct proof of [*!@] and offer that indefinitely large sizes must be indiscernible, otherwise, there would be little distinction between them and those of "ordinary size". "The indiscernibility is so much a part of one way in which we conceptualize indefinitely large sizes that we usually fail to distinguish different indefinitely large sizes in any way at all. Though we can easily see that indefinitely large size is context relative, we often act as if it were not" (pg. 256).
RP: Master do not forget about extrapolation from [*!@]
Lavine: Yes. One important reason for taking an extrapolation is implicit from finite mathematics itself. In finite mathematics we can extrapolate from everything.
FE: It seems that you may be smuggling in out intuitive notion of infinity somewhere in this argument Lavine.
Lavine: Trust me, I assure you infinity can be understood. Where was I…? Another important reason to extrapolate is that "we often regard the apparent indefiniteness of some quantity or other as the epistemic reflection of an objective fact" (pg. 266).
FE: I think I may be starting to understand you, this is dangerous. Where are the rest of my pills damn it!
Lavine: That last point I made also relates to the formulation of [*!@#] because the "axioms of set theory are extrapolations of evident principles concerning indefinitely large size" (pg. 251).
RP: It was incident to the formulation of indefinitely large in the origins of modern set theory.
Lavine: Exactly.
Frater Elijah swooned as the images before his eyes swirled and melded together. He thought he saw through the fabric reality, to a place where people actually took this stuff seriously. What a trip.
There was a screaming sound as the nothing erased a section of Lavines leg.
RP: Master, oh no!!! What has happened.
Hassan reformed from the surrounding space.
Hassan: Sorry Lavine, it looks like you are done for. Rishi P. There may still be hope for you.
Nobody: Nothing can escape me! HAHAHHAHAHHAHA!!!!!!!
Lavine: Help me somebody help, what of my theory…….
Lavine vanished.
FE: That is the key. Nobody said it!
Rishi P by a means of anti-logical deduction (induced from the instant enlightenment experience of possible death) realized, and understood. Nothing can escape. He remembered way back to his old Taoist master Lao-Tzu who taught Rishi P. about the infinite (also called Tao).
{fade to a brief memory sequence of a wise looking old man in robes}
Lao-Tzu: "The Tao that can be described is not the eternal Tao. The name that can be spoken
is not the eternal Name. The nameless is the boundary of Heaven and Earth. The named is the mother of creation. Freed from desire, you can see the hidden mystery. By having desire, you can only see what is visibly real. Yet mystery and reality emerge from the same source. This source is called darkness. Darkness born from darkness. The beginning of all understanding" (Tao Te Ching).
Frater Elijah understood this mystery a long time ago and faded from Nothing to Nothing with ease, Hassan obviously knew a way also. Rishi attempted this, just as the void approached and he realized that he had nowhere to go…
Hassan and Frater Elijah plummeted through a vortex forged from nothing towards existence. Both crashed into a pile of wood sticks. Somehow they have managed to arrive at the site of the burning of Joan d'arc. There was a bloodthirsty mob surrounding our trio, screaming for what blood-thirsty mobs love the most. When the light from the sun was blocked out.
A hush through the crowd
Screams!
Anarchy!
The crowd scattered!
But where was there to run?
Elijah looked up and was thankful for all of the pills he took in the nether regions. What he saw, was none other than a giant disembodied uterus descending from outer space.