Going to Oblivion


Many of us are familiar with Rayo's number - an incomputable number that sails past anything derivable by BEAF. It is usually defined as:

The smallest integer larger than the largest finite number defined by an expression in the language of first order set theory with a googol symbols or less.

The Rayo function R(n) is usually defined like so:

The smallest integer larger than the largest finite number defined by an expression in the language of first order set theory with n symbols or less.

I became interesting in coining a new number, MUCH larger than Rayo - I even had the name picked out.

So the question is: how do we beat Rayo's number? One way to do this would be to do recursion on the Rayo function, such as R(Rayo), R(R(Rayo)), R(R(R(R(R(......Rayo times........(R(Rayo))).....)))). As powerful as this seems, it would only add 1 to the speed Q on the FGH (Fast Growing Hierarchy), where Q is the speed of Rayo's function - keep in mind Q is FAR FAR FAR larger than w1 an uncountable ordinal, so adding even the Large Veblen Ordinal (MUCH larger than 1) is like adding an atom to a kungulus universes - not very effective! So now what do we do - how about doing BEAF style recursion on the Rayo function, well this is better, but still not enough. Adding q(Ww) (the ordinal power of legend arrays) to the speed will not make a significant budge to Q. Even Rayo function style recursion over and over again would be no more effective than adding only one entry to an array capable of keeping up with Rayo. OK - NOW what? Lets look at the definition again to Rayo's function, but I'll highlight something.

R(n) = The smallest integer larger than the largest finite number defined by an expression in the language of first order set theory with n symbols or less.

So what if we used second order set theory, or Rayo order set theory - heck what about LVO order set theory (assuming this is definable). We could also define a Rayo like function that looks for the largest ordinal within a set number of symbols and use that ordinal for a kth order set theory and so forth. I attempted to come up with a recursive like system to define my new number, but it was looking a bit too unwieldy, I wanted something simpler to define, but far more powerful.

Later Tyler Zahnke told me about a new number called BIG FOOT invented by LittlePeng9, which uses a more powerful theory than set theory called first order oodle theory (FOOT), it was a well defined system that could deal with higher order set theories and blast past anything that could be described by what I written above. The FOOT simply added a couple new symbols [] and descibed how they functioned and something called oodles.

I was about to drop my number idea, but then I thought of something that might knock the socks off BIG FOOT - I just hope it stops. Both Rayo's and BIG FOOT uses a simple definition that are almost the same, but where BIG FOOT uses FOOT theory instead of set theory. Both theories can be completely defined within a few kilobytes of info. So lets define a K(n) system. A K(n) system is a complete and well defined system of mathematics that can be described with no more than n symbols. I suspect that both set theory and FOOT would be a K(10000) system (not sure how many symbols would be needed exactly, but both can be defined completely on a web page. Now for my number - huhuhuhahahaha!

I define Oblivion as the largest finite number that can be uniquely defined using no more than a kungulus symbols in some K(gongulus) system.

Considering the '10' in the definition of BIG FOOT - i.e. BIG FOOT = FOOT10(10100), I wasn't sure how powerful it was - was is something along the lines of - start with a K(10000) system then find a maximum number MK(10000) then use a K(MK(10000)) system and repeat it 10 times?, I wasn't sure. If it does this, then BIG FOOT would beat oblivion - so I decided to come up with another number.

Lets let a K2(n) 2-system be a complete and well defined 2-system with up to n symbols that can generate K systems and have a largest finite number as long as n is finite, the generated K systems have finite number of symbols and the number of symbols of the language written under the K system is finite - example: we could define the largest finite number uniquely defined using a kungulus symbols in some K(gongulus) system within a K2(gongulus) 2-system. The K-systems are like first order mathematical systems, while the K2 systems are like second-order systems, but even some K-systems can be defined to beat higher order smaller systems - i.e. FOOT beats LVO-order set theory.

We can continue with K3 3-systems that can generate K2 2-systems and so forth. Now lets also allow the number oblivion to be repesented with one symbol.

Lets define Utter Oblivion as the largest finite number that can be uniquely defined using no more than an oblivion symbols in some K(oblivion) system in some K2(oblivion) 2-system in some K3(oblivion) 3-system in some K4(oblivion) 4-system in some .........KOblivion(Oblivion) Oblivion-system where the number oblivion can be represented with one symbol (byte). YIIIIKKKEESSS!!!

Feb. 3, 2016 addition: LittlePeng9 just revealed to me that the 10 is simply functional exponentiation, so BIG FOOT is FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(googol)))))))))). This means that Oblivion is larger. I also changed the word "byte" to symbol for a byte is 8 bits, which would limit the number of types of symbols to 256 which wasn't my intention - my intention was to use as many types of symbols as you wished.


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