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I have something new today!

And it's big.....It's a whole new theory that could construct very,very large numbers,possibly larger than the current record holder BIG FOOT!

It's MBOT (Mush9-Boris Output Theory).

Background

Ok,a while ago Mush9 wrote a blog post about the NOOP FUNCTION,which got me really interasted.

This could possibly beat the largest of numbers ever defined and change the way we think about the creation of numbers.There was no information for the function itself,how it works or what it actualy does.

So I decided to make one myself! On Mush's talk page I wrote a few messages about my idea.It would be very helpfull if you read them before reading this blog post.

Introduction

MBOT is based off ZF or ZFC.

The thing that is different than in set theory or any other particular Mathematical theory that I can think of right now is that in MBOT we can add systems which can recreate almost any existing function,given enough space.

So the first thing we need is to know what an operation type is.

In MBOT there are 6 types of operations:Statements,Theorems,Calculations,Functions,Systems and Notators.

For now I will focus on systems.

A system is the way that a function works:what gives it it's growth.

In MBOT we use \(\Phi()\) to denote this is a system-type.

Statements

Statements are one of the six main operation types in MBOT.A statement can be any information that would only work for sertain conditions.Statements are very similar to theorems,but do not need to be proven and would not work for every case but only with the given possibilities.An example of this could be anything.We can apply logic and use theorems to check if a statement is correct.If we have a machine that answeres questions with "yes" or "no",then in order for that machine to work,we have to use axioms,parts of theorems,and other types of theorms to confirm if one claim is true or not.Everything that we say to that machine would be a statement.The next logical step after that is to include the possibility,that there would be a question or a statement,that cannot be proven in MBOT.This would require the machine to have a third option,saying "Independent of MBOT".This whole machine thing would require theorems to exist in order for it to work.

Theorems

rder for it to work.Theorems are statements that are furthur developed to include the langiuge of MBOT to work.In order for a ststement to become a theorem,it has to be proven.After it has been proven(using the langiuge of MBOT to do so)that it would work in every condition satisfying the set conditions,the statement becomes a theorem and is now allowed to be used to check whether other statements are correct or not and the cicle continues......

For all of this to work we have to apply MBOT languige for it to work.

So what is this MBOT languige?

Well,MBOT uses the exact same formulas as FOST.That is the languige of MBOT is the same as that of FOST.Here the logical connectivities \((\forall,\exists,\land,\lor,\neg,\Rightarrow,\Leftrightarrow,(,))\) do have their traditional meanings.

Calculations

Calculations are the third operation type in MBOT.

Ones we have a certain statement and we have a theorem to prove it,we go furthur by trying to calculate the statement,using the theorem that coresponds to what the calculation is about.We use the theorem as a rule for the way we calculate it so we say whether the statement is true or not.If the theorem is used correctly,and the calculation is done correctly and the overall answer does not lead to contradictions,then the statement is proven true.

From here we build our MBOT machine.Here are the comands:

  • If there is a statement given:
  1. Find the theorem that implies the same case.
  2. Do the calculation using the theorem as a base case rule.
  3. Give the answer for the following cases
a) If step 1 is done correctly and step 2 is done correctly and it does not lead to a contradiction,then the statement is true.
b)If step 1 cannot be done because a theorem cannot be found for that case,then the statement is independent of MBOT.
c)If step 2 cannot be done because the calculation cannot be done for that case,then the statement is independent of MBOT.
d)If step 1 and 2 are done correctly,but is leads to a contradiction,then the statement is false.

And our machine is finally finished!

Statement                        Theorem used          Calculation               Answer
1+1=2 \(\forall a+b=c:c-a=b|c-b=a\)    \(2-1=1\) true
3*3=4 \(\forall a*b=c:{c \over a}=b|{c \over b}=a\) \({4 \over 3} \neq 3\) false
7>n for n<5 \(\forall a<b,b<c:a<c\) \(n<7\) true
n is transfinite for n+1= n \(\forall n+1 = n:n \geq \aleph_0\) \(n \geq \aleph_0\) true
Earth is round \(F_1 = F_2 = G \frac{m_1 \times m_2}{r^2}\) Too long(but it works) true
I am retarded [not found] [not possible]

Independent of MBOT(but probably true)                                                                                

V=4 for 1<V<13 [not possible]                                               [not possible] Independent of MBOT

Functions

A function is any \(\Delta_\kappa\) for which \(\exists \rho \Leftrightarrow \Delta_\kappa=\rho\),where \(\kappa\) is an input and \(\rho\) is an output.There exist functions that are provable in MBOT and functions that are independent of MBOT.A function,that's provable in MBOT is any function for which there can be placed a Phi-system for which it will give out precicely the same output \(\rho\) for precicely the same input \(\kappa\).Similarly a function that's independent of MBOT is any function for which such Phi-system does not exist.

\(\Phi\)(Phi) basics

First,\(\Phi()\) means the operation is a system type.

Here we use the logical connectivities from set theory.However,non of them have anything to do with their traditional meanings! This should not confuse people.

\(\rightarrow\) here means next step/character/move.

\(\cup\) means a long chain of repetitions.

\(\leftrightarrow\) means that the system returns _ if _.

\({_n}\),where n is a subscript means an n-level system.For those sequences to be infinite usualy they collapse in on themselves.

\(\sim\) on itself has no meaning and can be set to anything,even more than the others!

Now,you might be noticing a problem with those:There definition is very weak and ill-defined.

So we can have an equasion-like explanation of them in the actual system.

Since their definitions aren't well-defind that leaves them open for any definition that you can set,respecting the conditions above.This makes Phi systems particularly strong,since we do not need a new symbol for every type of recursion possible,but rather set the meaning of them in the system.

Next we can define more than one systems,that operate differently,but have one be based on the other.Then we can say one of them is inside the other.

Examples

These are some examples of Phi systems in action:

The Ackerman function(\(Ack(a,b)\)) would be a \(\Phi(a)\rightarrow b=\Phi(a-1)\rightarrow (\Phi(a)\rightarrow (b-1))\) system,inside of a \(\Phi(a)\rightarrow 0 = \Phi(a-1)\rightarrow 1\) -system,inside of a \(\Phi(0)\rightarrow b = b+1\) system.

What would four-entry arrays in BEAF(\(\{a,b,c,d\}\)) be?

It would be a...

\(\Phi(a) \rightarrow b \rightarrow c \rightarrow d = \Phi(a) \rightarrow (\Phi(a) \rightarrow (b-1) \rightarrow c \rightarrow d) \rightarrow (c-1) \rightarrow d\) -system,inside of a \(\Phi(a)\rightarrow 1 \ldots = a\) system inside of a \(\Phi(a)\rightarrow b\rightarrow 1\rightarrow d = \Phi(a)\rightarrow b\rightarrow (\Phi(a)\rightarrow (b-1) \rightarrow 1\rightarrow d)\Rightarrow (d-1)\) system inside a \(\Phi(a)\rightarrow b = a^b\) system.

If at some point the system changes,it will have to be pointed out that the system has changed and the primary system will be sent back inside the secondary system.

Example:What would \(f_3(n)\) in the fast-growing hierarchy be?

It would be \(\Phi_3(n)\) inside a \(\Phi_x(n) = \Phi^{n}_{(x-1)}(n)\) -system inside a \(\Phi_0(n) = n+1\) -system

Pretty easy....right?

Well at this level,yes!...

Now we can start using other symbols for stronger definitions:

Let's now get the same definition for the BEAF system and put it all inside of a

\(\Phi(a)\cup b = \Phi(a)\underbrace{\rightarrow a\rightarrow a\rightarrow \ldots \rightarrow a}_b\) -system.

This way \(\Phi(n)\cup n\) has the growth rate of \(f_{\omega^{\omega}}(n)\)!

Now some stronger definitions:

\(\rightarrow n\) means repeating everything previously set in the operation \(n\) times.

\(\rightarrow n\rightarrow m\) means repeating the system n,m times.

\(\rightarrow n\rightarrow m\rightarrow k\) means repeating the system \(n\rightarrow m\), k times.

\(a \cup 1\) means a long chain of repetitions

\(\Phi(a) \cup 1 = \Phi(a) \rightarrow (\Phi(a)\rightarrow a \rightarrow \ldots \rightarrow a)\rightarrow a\)

\(\Phi(a) \cup 2 = \Phi(a) \cup 1 \rightarrow \Phi(a-1) \cup 2) \rightarrow a\)

With just these operators we can set any system we want:

\(\Phi(a) \rightarrow b = \Phi^b(a) \rightarrow (\Phi(a) \rightarrow (b-1)) \cup a \rightarrow a\) -system for example is a system whith a growth rate of something like \(f_{\psi(\Omega^{\Omega^{\varepsilon_0}+b})}(a)\)

Another good system is

\(\Phi(a) \rightarrow b \cup c \rightarrow d = \Phi^b(a) \rightarrow (\Phi(a) \cup b) \rightarrow c \cup \Phi_c(b-1)\)

where \(\Phi_a(b) = \Phi_a(b-1) \rightarrow a \cup b\),where \(\Phi_a(1) = \Phi_{a-1}(\Phi(a\cup a))\rightarrow a\)

This has a growth rate of \(f_{\psi(\psi_I(0))}(a)\)

Now,It's time for uncomputable functions!

Uncomputable systems!

The problem with what we were doing previously is that no matter how long we make it,it will allways be computable and thus will be surpassed by the Busy beaver function,the Xi function,Rayo's function,FOOT function,......

So we now percieve the logical conectivity \(\leftrightarrow\) and the meaningless by itself \(\sim\).

We also use \(\mathbb{C}\) to denote if a system is computable,\(\mathbb{U}\) if it's uncomputable,\(\lor\) for minimum,\(^{>}\) for overgrowing,\(^{>^*}\) for eventual domination and \(^{>}+(@) F(n)\) for overgrowing another function/system F(n) by @ at a time.This means that the output of the described system gives an output \(\rho\),that is larger by @ than the output for for F(n).More precicely:

\(\forall F(n) = \rho ,F_2(n) \text{is a}  \Phi(n)^>+(@)F(n)  \text{-system} : F_2(n) = \rho + @\)

We can even manipulate it so that it doesn't just add it to the operators,but it uses higher-operations.

With this we can make uncomputable systems,stronger than all computable systems and now,just like we could make computable systems as strong as we like,we can now make uncomputable systems as strong as we like.

Now there is no limit to how far they go! Infact Phi systems are like word descriptions,but with symbols and thus any function that can be described on paper can be described in a Phi system.

That means that there exists a system using a finite amount of charecters that is larger than FOOT function.

I don't know how many symbols this requires but it should not be too much since the Busy beaver can be described in about 20 symbols:

\(\Phi(n) = n\sim n \leftrightarrow k \leftrightarrow m=\ldots 00\underbrace{111\ldots 111}_m00\ldots \leftrightarrow 0=\infty\)

This represents \(\Sigma(n)\),where the tape is infinite and set completely to zeros.

n represents the input,k represents the number of states and because of the \(\sim n\) part that becomes n and m is the longest possible line of ones that could be written using the other conditions and is the output.

My guess is that FOOT function should be able to be described in less than 1000 symbols,but I can't prove any of that without actualy showing the solution and I don't have it.

Conclusion

The whole aim of this theory was to show that there are ways of writing a (sort of) turing machine for all functions and that there are obvious outputs for most functions.However it also shows that sometimes there is no obvious output!

This is where NOOP function comes in!

This whole thing was inspired by many things:

  • Mush9's idea for NOOP function and the whole No Obvious OutPut thingy.
  • ZFC
  • Turing machines
  • Computation
  • FOOT
  • Unprovability and Uncomputability Theory 
  • Bowers' K-system

We define \(\Delta_x\) as the strongest possible system describable in \(x\) symbols or less(describing every number as one symbol and excluding () and Phi).

Now we define \(\Delta_x[n]\) as the largest number describable in n symbols or less in that system!

This is exactly K-systems,exept this time it's well-defined! The reason why Bower's function was ill-defind is because it did not use any fixed languige for it's definition.Mine has.....and it uses MBOT as it's fixed languige! 

Now we define our final function \(\text{NOOP(n)}\) as \(\Delta_n[n]\) !

My final number will be \(\text{NOOP}^{10}(10^{100})\) and I will call it BIG NOOB!!!  

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