Infinite_Numbers

Sbiis Saibian's

!!! FORBIDDEN LIST !!!

of Infinite Numbers

Introduction

            Googology is really the study of finite numbers. However there is an analogous "large number" competition that is sometimes played with large infinities. The problem is that this game is much more difficult to define any ground rules for and disagreements are even more likely to occur about which "number" is larger than in ordinary googology. This game of infinities we'll call transfinite googology. My website is primarily about finite googology. However there are some reasons to bring infinity up even in conversations about large finite numbers. For example, cantor's infinite ordinals work great as indexes to the family of functions known collectively as the fast-growing hierarchy. For this reason, a study, at least of the countably infinite ordinals, is helpful in the construction of recursive super structures which in turn allows us to define extremely large numbers that we wouldn't be able to otherwise. 

                That being said, I abhor the use of "infinity" as an attempt to skip finite numbers all together. Finite numbers, especially large ones, are very fascinating in their own right. The problem is that according to "bigger is better", all finite numbers lose ground when infinity enters the discussion. The problem I have with this, is that soon after introducing infinity the competition quickly devolves into an absolute confusion of well-foundedness. Is infinity and one larger than infinity or not? How far are we willing to take the platonic existence idea? When we start talking about things which are so remote from reality, and so difficult to make any sense of that they may well be incoherent, can we really claim platonic existence? If transfinite numbers can simultaneously be shown to be larger yet equal to another transfinite, then what are we to make of this? When we match up the even numbers with the positive integers to show the sets are equal, how is this any more valid than matching up evens with evens to show there are more positive integers? Infinity is more than a really big number. It has properties which no finite number has, no matter how large! Infinity and all it's ilk are categorically different than anything we would normally consider numbers! For this reason, I have a somewhat cautious attitude in regards to completed infinities. However it does seem that at least potential infinity is necessary for googology and mathematics in general to function properly. So I will present a list of infinities here. They will be listed in "size order", according to cantor's theory of transfinite ordinals and cardinals. 

                Although this can be thought of a continuation of the "large numbers list" after all the finite numbers, you must keep in mind that even the smallest infinity aleph-null is infinitely larger than the largest finite number on my large numbers list. Even the largest finite numbers are an aleph-null units away from aleph-null. If this seems paradoxical, that's because it is. Infinity violates properties we implicitly accept for finite quantities, such as being uneffected by having something removed! For this reason I consider infinities to be forbidden quantities that violate basic laws of numbers. So in a sense this list can not be a continuation in the proper sense. This list exists in a completely different category of numbers, and there is no "bridge" between the two worlds. Something is either finite or infinite. Something can't be "nearly infinite", or "mostly finite" in the strict sense. There is no "gradual" journey towards infinity. A journey to infinity MUST always be a finite one, otherwise we never actually reach it. The only way to infinity is to instantly "jump" to it. So let's jump into the deep end of large number ... the forbidden void of infinities ...

ℵ0

aleph-null

             Also known colloquially as infinity. In Cantor's theory of transfinites it is known as aleph-null when treated as a cardinal number, and omega when treated as an ordinal number. In an informal sense all of these concepts are the same, but there are important technical distinctions to be made. Infinity in calculus refers to a real quantity which increases without bound. It is not so much a number, as a way of expressing the behavior of a limit. omega refers to the order-type of the set of non-negative integers. Aleph-null on the other hand, is defined as the cardinality of the set of positive integers. In plain speak Aleph-null is the "number" of numbers. The problem with this is that the set of positive integers is suppose to represent all things that we might wish to count. It however, can not count itself. So is the "number" of numbers, even a number then? Cantor thought so. In some ways we can treat aleph-null as a number, in that we can compare it to other numbers and determine which is larger. Using the concept of one-to-one correspondence Cantor showed that we can rationally say that aleph-null is larger than any positive integer, even though the previously prevailing wisdom was that infinity was not a number and could not be compared in this way. But accepting this view leads to some mind bending anomalies. Using one-to-one correspondence we can show there are just as many even numbers, squares, cubes, etc. as there are positive integers, despite that fact that these are all subsets of the positive integers. This violates the principle that the "whole is always greater than any proper part of the whole". So aleph-null is a number such that a proper part of it is still just as large... baffling. When working with finite numbers we implicitly understand the exclusivity of "larger" vs. "equal". A number can not be both. Hence when one particular correspondence shows that one finite set has more than another finite set, we know that no correspondence can exist which shows they are equal. Not so with infinite sets! Even if we have a correspondence which shows one is larger than the other, it doesn't necessarily mean that a correspondence doesn't exist showing they are equal. In the Cantorian universe of cardinals, in order for an infinity to be truly larger than another infinity it must be shown that "there does not exist ANY one-to-one correspondences". Since there must be an infinite number of such correspondences, checking each one individually is not an option. It is necessary to come up with a proof which shows the impossibility of such a correspondence. It might be assumed that all infinities are essentially the same and can be put in one-to-one correspondence with each other. The amazing thing Cantor did however was to show that there were infinities which could not be put in one-to-one correspondence with aleph-null. Thus Cantor showed that there was not one infinity ... but an infinity of infinities ... (See aleph-one). This is Cantor's paradise, or nightmare, depending on your perspective.

w+1

omega and one

                This is the smallest ordinal number after "omega". Informally we can think of this as infinity plus one. One formulation of ordinals is to treat them as sets of all smaller ordinals. In order to say omega and one is "larger" than "omega" we define largeness to mean that one ordinal is larger than another if the smaller ordinal is included in the set of the larger. The set w+1 would be {w,0,1,2,3,...}. It would be composed of all the non-negative integers plus omega. Thus w+1 by this definition is larger. However, since the cardinality of every ordinal is represented by the cardinality of it's set, we can also show that in a sense w = w+1, since w={0,1,2,3,...} and w+1={w,0,1,2,...} we can pair off arguments as: {(0,w),(1,0),(2,1),(3,2),...}, which shows both sets have the same number of elements, even though w+1 includes one more. Confused? Basically there is two ways to look at comparison of infinities: the cardinal view and the ordinal view. By the ordinal view, omega and one is greater, by the cardinal view omega and omega plus one are the same thing. Cardinals don't play a large role in googology, but the countable ordinals do. So for our purposes the distinction between w and w+1 is important.

w+2

omega and two

                Just as we can extend large numbers arbitrarily, we can do the same with the ordinal w. Just think of "w" as a VERY large number. So we can add one to it, or two, or have ...

2w

two omega

2w+1

two omega and one

2w+2

two omega and two

3w

three omega

w2

omega squared

w2+1

omega squared and one

w2+w

omega squared and omega

w2+w+1

omega squared, omega and one

2w2

two omega squared

w3

omega cubed

ww

φ(w,0)

omega to the omega

ww^w

omega to the omega to the omega

ε0

φ(0,1)

epsilon-zero

                Cantor gave this ordinal the special name epsilon-zero. What is it? It's the smallest ordinal larger than any ordinal that can be "named" using addition, multiplication, and exponentiation with the symbol w. In other words:

e0 = lim{w,w^w,w^w^w,w^w^w^w,...}

                    In other other words, e0 = w^w^w^... where there is omega w's. It can be defined as the smallest ordinal "a", such that a=w^a. This implies that e0=w^e0. Weird. It's also equal to phi(0,1) in the Veblen fixed point hierarchy. 

                Informally you can think of it as an infinite power tower of infinities! It can also informally be called w^^w. 

                This ordinal is actually important for us as it represents the size of Jonathan Bowers' tetrational arrays. Not only does Bowers' tetrational arrays have an exact arity of epsilon-zero (a tetrational array is well defined as long as only a finite number of entries are greater than 1. The rest are all equal to 1 by default. If we use the ordinals to count all the entries in tetrational space, there is exactly epsilon-zero entries), but function epsilon-zero of the fast-growing hierarchy has an equivalent growth rate to tetrational arrays. Epsilon-zero also represents an important impasse. Up to this point the notation is fairly natural, and there is basically universal agreement on how to "name" ordinals less than epsilon-zero and how to determine which of any two such ordinals is larger. At epsilon-zero however we begin to run into problems. There are at least two different ways to continue naming ordinals, and certain expressions are difficult to interpret, such as w^^(w+1). The fact is, that we are forced to make certain choices about notation after this point, and none of them follows as naturally as it does up to epsilon-zero. There is however a widely excepted extension known as the Veblen hierarchy. Unfortunately this extension is radically different than Bowers' own extension to pentational ordinals and beyond. It is an open question how to convert from Bowers' ordinals to Veblen ordinals.

ε0+1

epsilon-zero and one

                What's so hard about continuing... just add one. Well of coarse we can always add one in the system of ordinals, just like we do with finite numbers (this suggests that there is NO largest ordinal, just like there is no largest integer). The problem isn't so much adding one. It's what happens as we continue further along...

w^(ε0+1) / w*ε0

omega to the epsilon-zero and one / omega epsilon-zero

                Here we encounter one of our first problems, though admittedly pretty minor. There is more than one possible ordinal notation we could use. In the first version we are building towards a stack of omega's, in the second a stack of epsilon-zeros. You'll see what I mean as we continue

w^w^(ε0+1) / ε0^w

omega to the omega to the epsilon-zero and one / epsilon-zero to the omega

                Despite the fact that we have at least two different ways to write ordinals after epsilon-zero, the good news is that it isn't too difficult up to this point to create a correspondence between them. That is, we can convert one notation to the other and thereby compare ordinals in both systems and determine which is larger. The key to this conversion is the definition e0=w^e0. Using this we can convert the first form into the second as follows:

w^w^(e0+1) = w^(w*w^e0) = (w^w^e0)^w = (w^e0)^w = e0^w

                This takes some getting used to, but w^e0 is merely e0, while w^(e0+1) > e0. In fact it's worse than that because e0 = w^e0 = w^w^e0 = w^w^w^e0 = ... etc.

ε0^ε0

epsilon-zero to the epsilon-zero

                This is a cool ordinal. This is epsilon-zero raised to the epsilon-zero. What is this in the Cantor normalized form? Let's see (remember e0=w^e0):

e0^e0 = (w^e0)^e0 = w^(e0*e0) = w^(w^e0*w^e0) = w^w^(2e0)

                Weird. Still, it seems like the ordinals after epsilon-zero are well behaved so far. What is the limit of extending e0 using exponentiation though...

ε1

φ(1,1)

epsilon-one

            Epsilon-one is the next big step in the Veblen fixed point hierarchy. Epsilon-one can be defined as the smallest ordinal larger than any ordinal expressible using only addition,multiplication, and exponentiation on the ordinals "w" and "e0". One way to define it is as:

e1 = lim{e0+1,w^(e0+1),w^w^(e0+1),w^w^w^(e0+1),...}

            Now that you've seen this you can probably guess what happens next...

ε2

φ(2,1)

epsilon-two

            Epsilon-two is the limit of expressions using w,e0 and e1:

e2 = lim{e1+1,w^(e1+1),w^w^(e1+1),...}

εw

φ(w,1)

epsilon-omega

                Now that we have established a general rule we can continue to any ordinal index of epsilon ... including infinite ordinals. YIKES ...

εw^w

φ(w^w,1)

epsilon-omega-to-the-omega

εe0

φ(φ(0,1),1)

epsilon-epsilon-zero

εe(e0)

φ(φ(φ(0,1),1),1)

epsilon-epsilon-epsilon-zero

εe(e(e(...

φ(0,2)

Epsilon Limit Ordinal / zeta-minor-naught

                Here we reach the limit of the idea of "Epsilon-numbers". This ordinal is sometimes referred to as zeta-naught. However I've avoided using this notation here, because I have reserved zeta-naught for the hypothetical ordinal w^^^w. It is doubtful that this ordinal is that large. Still this is a pretty cool ordinal. It's usually the largest ordinal mentioned in a popular discussion of Cantor's transfinite ordinals. That's probably because after this, we need to develop a more generalized means for continuing and it becomes far less natural and more technical. Most authors would consider this sufficient as an introduction to Cantor's ordinals. After this it starts to get somewhat academic...

φ(1,2)

zeta-minor-one

φ(0,3)

φ(0,w)

φ(0,φ(0,1))

φ(0,φ(0,φ(0,1)))

Γ0

φ(0,0,1)

Feferman-Schutte Ordinal

            This is the Feferman-Schutte ordinal, also known as gamma-naught. It's the limit of the binary Veblen function. It is said that the growth rate of the function TREE(n) is comparable to gamma-naught of the fast growing hierarchy. Determining where gamma-naught falls along Bowers' ordinals is therefore important to determining where TREE(3), a famous large number, falls along Bowers' googolisms.

φ( w & a )

Small Veblen Ordinal

            The small Veblen Ordinal is the limit of extending the Veblen function to a Veblen array. 

φ( w & a & a )

            It is possible to extend Veblen arrays, just as Bowers' extends his array notation. This is an example of Bowers' idea in professional mathematics!

φ( ... a & a & a & a )

Large Veblen Ordinal

            The Large Veblen Ordinal is the limit of allowing the arity of an extended-Veblen array to be any ordinal definable using an extended-Veblen array. Despite how huge this is, it is still a computable ordinal. That means that any ordinal less than this in the fast-growing hierarchy is still a computable function.

w1ck

Church-Kleene Ordinal

            This is the infamous Church-Kleene Ordinal, said to be the smallest non-recursive ordinal. Strangely, even though it's non-recursive it's countable. This means that the cardinality of the set w1^ck is still aleph-one! It's like we haven't gone anywhere yet, even though we have gone a very long way in terms of ordinals. The distance between this ordinal and the Large Veblen Ordinal is insurmountably vast. 

            This ordinal is also equivalent to the growth rate of the Busy Beaver function.

w2ck

            One can also have a hierarchy of non-recursive ordinals.

w(w)ck

            A.P. Goucher has claimed that this is the growth rate of the function Rayo(n).

w(w(w( ...|ck

First Church-Kleene Fixed Point

        This is the growth rate of A.P. Goucher's Xi function. Currently the fastest growing function in googology.

ℵ1

aleph-one

            Aleph-one is the first uncountable number. What does that mean? It means that any set with at least this many elements can not be put into one-to-one correspondence with the set of positive integers. Put another way, it's a number so large that even if you attempted to list it's arguments 1st,2nd,3rd, ... and continued this list forever, you still wouldn't be able to account for all it's arguments!! This number is larger than infinity to its own power, larger than a power tower of infinities infinitely high ... it simply defies description. It can't be compared with aleph-null. At first it seems impossible, nightmarish. Yet it's existence seems inescapable. Cantor did not set out trying to prove there was a larger infinity. He stumbled upon it by accident. He assumed that all infinities were the same. But when he tried to set up a one-to-one correspondence between the positive integers and the real numbers he discovered he couldn't. Eventually he realized that the reason he was having so much difficult had to be because no such correspondence actually existed. So he then set out to prove there were more reals than positive integers. Here he succeeded. It was soon also discovered that the power set of aleph-null had to be greater than aleph-null. Cantor tried to prove that the power set of aleph-null was the same as the cardinality of the real numbers, but he was unable to do this. Some say it was this question that drove him mad. Later on it was proved that the question itself was undecidable, meaning no proof existed within Cantors framework that would show that the power set of aleph-null was also the cardinality of the reals. It's at this point that we learn something disturbing about studying infinity: that certain "truths" about them are unknowable and must simply be assumed one way or another. Some mathematicians have assumed Cantor's conjecture to be correct, or at least a useful way of working with infinities, and have taken it as an axiom, where others have explored other possible systems. But this begs the question: how can infinities defined in radically different systems be compared. The most important feature of any large number competition is that eligible entries must be well-ordered. That is, for any two distinct members, a larger member can be determined, at least in principle. When that breaks down we no longer have a competition, and for me personally, that is the essence of googology: a competition with a verifiable champion.

N2

aleph-two

            The next hypothetical aleph number, which can not be put into one-to-one correspondence with aleph-one or lower. Again, the power set of aleph-one is also larger, but we can't determine whether the power set of aleph-one or aleph-two is larger.

            This generalization of coarse leads us to...

Nw

aleph-omega

Ne0

aleph-epsilon-zero

NN(1)

aleph-aleph-one

NN(N(1))

aleph-aleph-aleph-one

N(N(N(...

First Aleph-fixed point

Ω

ABSOLUTE INFINITY

                This number is a FORBIDDEN NUMBER above and beyond all forbidden numbers. To Georg Cantor, who was religious, this number literally represented the totally boundless nature of GOD's power! The concept of Absolute Infinity is so bizarre that it isn't even considered an official transfinite number! This is because the concept itself is a nightmare of contradictions and incomprehensibility. It serves virtually no purpose in mathematics except to mystify the mind of man. It's an infinity amongst infinities, in a very literal sense. Absolute infinity can be thought of as the limit of ALL TRANSFINITE NUMBERS! How is this even possible? Cantor defined absolute infinity by saying that any property you could imagine it being the first to possess, is already possessed by a lesser number. What this means is that we literally can not describe anything about this number. Any description would be referring to something other than absolute infinity! Sound familiar? It's exactly the same thing that philosopher's have said about GOD. That we can't say what GOD is, only what GOD isn't. But how can this make any sense?! If we empty a term of all meaning, what meaning could it have. Absolute infinity actually already contradicts itself, because by Cantor's own logic, the power set of any set is a larger set. But then the power set of Absolute infinity would be larger, but then it wouldn't be larger than all transfinite numbers!!!

                Cantor's vision of the infinite is mind-blowing, and bizarre almost beyond description. In Cantor's own time his ideas brought on much ire from his fellow mathematicians. Yet not everyone wanted to silence these new ideas. It was described by Hilbert as "a paradise from which we will never be evicted". But Cantor's infinities are a lot like pandora's box. Once we open it, there doesn't seem to really be an end whatsoever. Where as before we could call infinity the largest number possible, now even that isn't enough. Yet here at absolute infinity we see the same human impulse that lead to infinity in the first place. The desire to declare a stopping point. A desire to comprehend the incomprehensible and give it a name. A desire to bring a close to things which are naturally open-ended. This is in contradistinction to the human impulse to move forward unimpeded. With absolute infinity we see Cantor being contradictory by trying to hold on to both ideas at once. Is it such a stretch to suggest that maybe old plain vanilla infinity might also be such a contradiction? How can any number claim to be the largest and final number? But if we gain nothing by declaring infinity the largest, then maybe we should just embrace the boundless. We don't need a last number. We think that's what we want, but three seconds later we would just define a larger one. So let's just admit, there is no such thing as a largest number, and that's okay. We don't need to name all the numbers today, tomorrow or ever, ... the numbers will always be out there waiting patiently. There is no rush............................................................................................................................................