That's called the successor function, it's basic, and all a mathematician needs to create bigger and bigger integers.
You can do it on your fingers too. First you take one finger, then two, then three, ... up to ten, and you've reached your ambidextrous limit. They say some people have six fingers on each hand, twelve fingers in all, but I've never seen them.

Ten evil deeds is a lot, I bet most of us would feel sorry for the rest of our lives. Anyway, betting is wrong, we must have certainty, certainty and big numbers, certainty that our number will be the biggest in the whole goddamn chiliocosm.
We need better methods than counting 1 + 1 = 2, be it on fingers or in breaths. I say we start counting the counting itself, can we do that? Yes we can! It's called multiplication.
Instead of counting with 1 we count with the results of our previous additions: the integer numbers, to create... bigger integer numbers (with less effort).

I'm a computer programmer, I use stars * for multiplication, and arrow heads ^ for powers. Also I must remark the "a" is just an abstract name for a specific number: a variable. You can store anything in it, as long as it can be part of the equation, as long as it is "some number".

Going from multiplication to powers is very easy: you count, you count the number of times you multiply by a number:

Something funny happened here, you might not have noticed because you're so used to it, but I actually started using a number system to write down numbers. It's the 216, see?
With the introduction of a more succinct way of writing down (and talking about) numbers, we don't need a different symbol (or new word) for each integer.

Given a certain base (in our case decimal base), we combine addition, multiplication and powers in a systematic way so to uniquely express any number.
Thought I'd write it down for you mathematically, see how smart you are that you do this all the time, 'cause what you do in a given number system (or radix, radix 10) is actually this:

2007 is a 4 digit number, counting the year we're in since the birth of our friend Jesus. Within the computer another reference point is kept to start counting time though. On the 1st of January 1970, the Unix Era started. Your browser measures it in milliseconds, currently how many is shown when you place your mouse cursor over the word Unix Era.

If we leave the confines of our room we can count the years to the creation of earth and the sun in 10 decimal digits, 11 decimals to the very start of the Big Bang, 24 decimals to describe the number of stars in the universe, and about 80 decimals to exactly pinpoint the number of particles it contains – if you could come up with a better way of counting that which remains shrouded in dark matter.

Yes, this number system seems enough for most natural purposes, thank you, we are done (perhaps primes are more of a mystery here than if we'd just stick to rearranging fingers, there's a downside ;-) or...
Can it be worthwhile to delve into even higher numbers? Numbers that can't be expressed in the familiar decimal system, but with operators whose construction method you're already familiar with?

This introduced the multiple arrow head operator, in this notation you obtain a lot of decimals from just two tiny numbers. Just what we need in our zest for magnitude.

I'm not going to beat around the bush, you can apply the notion of raising super powers to triple heads ^^^, quadruple heads ^^^^, etcetera. Each next arrow head expanding the use of its predecessor in the same manner as ^^ expands the notion of ^.
That's not difficult: your counting again kiddo, counting arrow heads now...

In fact I feel rather silly using such an awkward symbol system for my operators, pretending to reinvent the wheel after every mile. We just need new tires: a way of counting operators, starting with addition as 1.

An operator function O, call it "bigO", naturally comes to mind. I've chosen to put the operator counter as a postfix to the numbers.

You're still with me? Let's continue our ascent into number asylum heaven.
Next step in the bigO curriculum is big Man arithmetic.

Enters the Anchorman numbers A.
These look like Ackermann numbers, but are anchored in addition instead, so I call them after my superhero Anchorman from Anchorage, agreed?

The Anchorman numbers suggest a way to specify what the next bigO operator function O(a,b,c,d) with 4 arguments (instead of 3) means:

If the fourth argument is 0 or missing, you'd use the regular numbers n in the equations.
But if the fourth argument has the value of 1, it means that you must substitute Anchorman numbers A_n for the regular numbers n in the equations.

And if the 4th argument has a value of 2, you'd use Barman numbers B_n.
And if the 4th argument has a value of 3, you'd use Caveman numbers C_n.
And if the 4th argument has a value of 4, you'd use Doorman numbers D_n.
And if the 4th argument has a value of 5, you'd use Echoman numbers E_n to substitute the regular numbers n in the equations.

Am I repeating myself? Who are these men?
Each next big Man in my alphabet is defined by:

So if we now write O(9,9,9,9), nobody has a clue how big this is. This is beyond imagination.
Yet you may call him the second (coming of) Anchorman number 9, and conveniently note him down as ₂A₉, and I'll know who he is, that number: he substitutes all occassions of 9 in O(9,9,9) with first Indiaman numbers I₉, which is the super power I₉^^^^^^^I₉ and makes me fear Indiaman most.
I can represent Indiaman as a Hotelman but that doesn't decrease my worry: I₉ = O(H₉, H₉, H₉) shows clearly that his H₉ - 2 arrow heads is a power this fellow cannot raise; it's the decisive maximizer that escapes old notation.

I must have lost you somewhere above, reader man, but hang on to your teeth now, there's even bigger news under way. As always we'll generalize past results into a broader approach, that's how we make progression...

This defines bigO completely.
We've run out of numbers, we've run out of operators.
The only pain we've left now is the length of the list of terms we need to write down for this operator function to work. As these are countable, they are calculable, we can define them by numbers. Of course we want only the best...

We need some help of what I call the little Sisters. The big Men did the hard work, now it's time for the little Sisters to wrap it all up. And make no mistake about the little Sisters, they got big boobs (just redirecting internet traffic :o)

The first Sister is most modest, I call her Antwoman or α
Similar to the initial big Man, she counts terms, but then she fills in that number across the board.

Now Bearwoman β muscles in and puts an α over α number of terms in the list of arguments to bigO. Bearwoman number 2 for example is a list of terms the size of α_α2 which is Antwoman number 4, and that's so huge!
I've worked out the operator steps you need to take to get α₄ on the blackboard above.

After Sister numbers 1 there is no more hope. We cannot get a clear picture anymore of what operators are involved in creating our numbers. What remains is Sister madness.

Catwoman crawls in the γ number basket and fills bigO with a repetition of β over β over ... whose sequence size is equal to that of the β_γ in question.

Dogwoman δ delivers Sister iteration dimensions you have never dreamt of...
Elephantwoman ε remembers recursions way beyond my comprehension...
Flamingowoman laughs at all you discrete number mathematicians with your fancy Greek letters, and...
Goatwoman catches the Gödel number I am sure. But then it's too late...

Haremwomen take over! More boobs.

Suppose I studied combinatorics and recursion for a year and then come back to this subject to create even better approaches or algorithms. Find out what Dogwoman was about, and continue further in her trail...
Suppose I got real smart and took another year to program a fast algorithm creation algorithm for big numbers, and that it could keep tabs on the notation of what all these algorithms amount to, and which one is best.
Suppose we let such a computer run for a year. Or make it a distributed computing project, like the one SETI sends out on the internet to catch the aliens. Or better still: buy a quantum computer from the aliens once SETI lured them into a trade agreement.
Could those algorithms ever reach a highest number?

And if we succeed in counting algorithms, what comes after algorithms?
Can we come close to the edge? Close enough to proclaim the end of countable numbers?

I won't mention the possibility of enumerating hypothetical years of computer work – oops I did, how bad can you get?
Suppose the big black Anti-Buddha, before eventually reaching enlightenment, as we all know, has to reincarnate a Catwoman γ₉ number of years. And that for each of those reincarnations he'd have to delude a Dogwoman δ₉ number of souls, who in their turn will get involved with more women and computers and 999...

It would be like an old Beatles record that got stuck for ages in the last groove. It would be bad indeed Subhuti.

So what does the World-Honoured One do?

The good Buddha multiplies by zero, and all our numbers vanish, as though nothing was done.
Multiplying by zero is such a powerful trick, and everybody can do it. Every day!

Does somebody have a good idea? More work to do? Windows of opportunities?
Don't add, don't collect. Multiply by zero, and nothing remains.

It's the bane of the Anti-Buddhas, mildness, like it says in the poem, "When the tree falls down, monkeys scatter everywhere."
That's good news in our cocaine ridden days.

Now I believe there must be a poison by which to counter a number so mild as zero. Maybe it is to be found in the higher infinite of cardinal arithmetic, that starts with the number ω.
ω is no longer countable, and as such isn't based on 1. It is a conjecture that this number is the first point an integer can never reach, because ω defines the total of the set of integer numbers.
Such definition is frowned upon by those who think numbers must be constructed to have any reality to them, but the field of set theory which occupies itself with infinite numbers produces the most wonderful mathematical theory, so there must be some truth up there...

If you've followed our construction of ever higher integers above, you'll appreciate that even carefully constructed numbers become more and more diffuse the higher we get.
First the numbers themselves are lost out of sight. Then the operators at the origin of bigO begin to blur. The self-iterating counters of arguments to bigO lose track. And maybe if we can count the algorithms which produce large numbers their reputations will vanish too.

Akihiro Kanamori wrote a truly great mathematical book "The Higher Infinite", from its 1997 edition I will give you the chart of names in the hierarchy of cardinals. A higher number here means a probably uncountably bigger set.
Those names already are like the poetry of mathematics. First comes ω, which is only weakly inaccessible.

That final 0 = 1 you can try to understand best. When a mathematician derives 0 = 1, one of the premisses or axioms which gave rise to such a conclusion is considered false. The trick in set theory is then to discard an axiom and try again.

0 = 1 is the gate-keeper of mathematics, the final word.

As such it is debatable whether it should be part of this list of cardinals. But its position as the keeper of the mathematical superstructure is not open for debate.

If Nansen would have wanted to kill the mathematical cat, he would surely proclaim 0 = 1 as the new axiom.
After that no number will hold anymore. All becomes interchangeable. This is equally true for the number 1 as for all countable integers as for the higher infinite numbers.

No room for the Anti-Buddha here anymore. Where can he go?

Anti-Buddha went down into the Netherworld and sought out all the great mathematicians, logicians and philosophers of all time. He locked them up in a small room, turned on the heater and made them think:

"The Buddha multiplies by nought and gets rid of all the integers, Nansen kills the cat for all cardinal numbers, what can we do? What options are left?"

It was Aristotle, who had been talking at the back end of the coffee table with the 20th century Wittgenstein for quite a while. It was him, the founding father of logic, first among Greeks, who came up with the solution to banish 0 = 1 from the system.

Aha! Anti-Buddha knew what he had to do, this was like throwing corn on the windmills of his mind. Why did he mess with equality all his precious life, who needs being equal?
Implication, causation, that's the answer. Understand problems, create solutions, achieve results, do something about it, start a political carreer, get the job done!

From the ground comes number one, from one two and the rest of the numbers. Does anybody want to look back to the mud? Nobody does! Equality was never an issue, something cannot reasonably be something else, it is absurd, unphysical.

Anti-Buddha, calling himself A→B by name now, as he considers himself the root of causation, makes an emotional appeal. He chooses existence, he chooses the one (the +1→) with its knots of numbers, he chooses the flux of time from which nothing can come back. He hates the Buddha's nirwana, the original face of Zen, the cessation of worldly bonds by Catholic saints; peace, simplicity, poverty... In A→B's philosophy movement and change are the key words, to spend it all in a spree and cry out loud: "Faster Pussycat! Kill! Kill!"

Anything goes? Anything goes fast? And it kills?
I started to worry, perhaps for the first time...

A logical test. We write a proof on the blackboard, that in "oneness", once you've started counting numbers on the road of no return, truth and logic don't mean a thing, because all becomes true.
Logicians need 0 as their false in the machine, that multiplies to 'and' two propositional variables A&B.

In "oneness" all is true and therefore logic ceases to function. Now that's what I call putting Joshu's sandals on your head! And it summarizes the position we have in the physical world. Because were it not for the recycling of our materials through death, we would be lost forever in tautology, and the only lie we'd be sure of was a faked one.

As I understand Wittgenstein he stated just this in his Tractatus Logico-Philosophicus:

The world is everything that is the case... The world falls apart into facts... Connections are independent of one another...
When a sign is not used  [in our example 0 or false]  it has no meaning.

Of course, without logic the dark artist who changed his name to a symbol (A→B, the implication) can no longer rely on the function of that logical operator to take him wherever he pleases. Because due to the wild assumptions he made in the past (namely "oneness without flaw", the flaw being zero), either logic itself has fallen apart (right after spitting out its death message on the blackboard) or his own special breed of "oneness" has been proven wrecked from the start.

Anti-Buddha we call him again, is beaten in arithmetic, set theory and logic. No place for him here. I guess he isn't taken by surprise. His royal badness was always more active mingling on the ethical plane. If there's still anybody out there...

Last time I left from the Zen River centre in Uithuizen, closing the door behind me on my way home, it was as if I heard Zen master Tenkei say, softly, "there is one more thing..."
It struck me like a koan, some Zen riddle, Tenkei's one last thing, you might call it, and he could become famous for it, I am sure, with everybody thinking: "How come Tenkei has a last thing?"
Other Zen masters might send their monks to the Zen River to finish their koan study with Tenkei Sensei, covertly asking their pupil when he came back, "What is that last thing of Tenkei, tell me?" Of course the mystery becomes greater and greater as the years pass by. Then finally on his death bed, when the master is old and deaf, one student gets approval. Now someone else knows, Tenkei's final thing, ah...!

This lengthy textual exposition is finished now, and I'm glad that it does have an end. You may read it as an introduction to Tenkei's one last thing – the phantasm which came over me as I closed the Zen River temple gate.
I know that koan study is a personal investigation in your own soul, and I never actually engaged myself in such training, but anyway, here you have some clues about the scientific meaning of the word "last", and how it sprouts from "one".
The essay offers some discoveries I made, innovative -yet natural- mathematical theory, like the recursive operator function bigO, that pleased me most. I hope it all helps.
Fran6 van Novaloka - September 2007