welcome to the big number battle where the biggest number wins in this game players will alternate writing numbers on the board ensuring that each number surpasses the preceding one the game concludes when a player is unable to produce a larger number in the blue Corner we have Augustine Rayo a philosopher of logic metaphysics and language in the Red Corner we have Adam elar a philosopher of science decision Theory and epistemology Rayo will begin the game let's play our first number on the board is the iconic digit one alga counters with seven ones to create 1, 111,111 Rayo responds with a genius move of changing alga's number into 11 factorial factorial factorial factorial factorial in order to understand the incredible magnitude of this number we must first understand what a double factorial is 11 double factorial is equal to 10,395 this is because 11 is an odd number meaning the double factorial of 11 can be calculated by multiplying every odd number up to 11 so what is 11 factorial factorial factorial factorial factorial we can rewrite it as 11 double factorial double factorial factorial we know 11 double factorial equal 10,395 so we can substitute this in now we have 10,395 double factorial factorial we can calculate 10,395 double factorial since 10,395 is odd we must multiply every odd number up to 10,395 which results in this astronomically large number with 18,6 123 digits finally we must take the factorial of this number which is going to result in a number with this many digits alga knows he has to go big now so he pulls out the Busy Beaver function of a Google Let Me Explain how how monumentally large this number is imagine you have an infinitely long piece of tape with an infinite number of zeros attached to it now imagine we put a busy beaver at the center of the tape the beaver needs to know what we want it to do so we give it a card with instructions the first number on the card asks the beaver to check if the number it is on is a zero or a one if the number is a zero the beaver will follow these instructions and if the number is a one the beaver will follow these instructions the second number tells the beaver what number he must swap with the number on the tape if the number on the tape is the same as the number he is told to swap it with then the number will remain the same the third number tells the beaver whether it must move left or right along the tape after making the swap zero means he must move one space to the left and one means he must move one space to the right finally the fourth number tells the be what instruction card to look at next for his next instructions let's look at a simple example of this with a single card first the beaver checks to see what number is on the tape he sees that it is a zero so he will follow these instructions next he sees that he must Swap this number with a one he then sees that he must move left because the third number on the card is a zero finally he sees that his next instructions are on card one which is the card that he already has so he repeats this again check if number is 0 or one swap Zer with one move one space to the left go to card one this will continue an infinite amount of times creating an infinitely long string of ones let's look at another example in this example the beaver sees a zero on the tape so he goes to the top of the instructions once again he then sees a one so he changes the zero with a one following this he sees the third number on the card to be a one so he moves right finally the fourth number on the card is zero so he goes to card Z card zero is known as a Halk card as it means the beaver must stop so this single card example simply created a one in the middle of the tape by adding more instruction cards for our Beaver to follow we can increase the complexity of the strings of ones that we can create the value that we input into our Busy Beaver function is the number of instruction in cards that we are using for example The Busy Beaver 3 function would use three instruction cards the result of each function is the maximum number of consecutive ones that can be created by a specific number of cards for example if we are only using one card then the largest finite number of consecutive ones that we can produce is one so the Busy Beaver function of one is one here are some values of some other Busy Beaver functions you will see that there's a question mark on the value for Buy Beaver function of five this is because the computer program that is required to calculate this value is still running you may also notice that we don't actually know the value of busy beaver function of six and this is because we don't yet have sufficient computing power to make such calculations in a reasonable amount of time the Busy Beaver function grows faster than any other computable function so just try and imagine how astronomically massive The Busy Beaver function of Google is Rayo has to come up with something amazing to beat El's number and that is exactly what he does with reo's number in order to get an understanding of what rayo's number is you must first understand the basics of first order set theory language first order set theory is a formal language used in mathematics to clearly Express mathematical Concepts it includes symbols such as the one seen here with each symbol having a specific specific function in the language this first order set theory expression is equivalent to zero this is because the first two symbols state that there exists a set called X the next two symbols state that for all elements Y in set X and the symbols inside the parentheses state that each element Y is not an element of set X this results in an empty set which is equivalent to zero in first order set theory now let's look at how this relates to rayo's number rayo's number is defined in words as the smallest number bigger than any finite number named by an expression in the language of first order set theory in a Google symbols or less basically what this is saying is that if we take a Google first order set theory symbols and we use these to create the biggest number possible rayo's number is the number just bigger than this however however there's a problem with this definition the problem arises from a paradox known as Berry's Paradox have you ever heard the question what is the smallest positive integer not definable in under 15 words in order to answer this question we could let X be equal to the smallest positive integer not definable in under 15 words but in doing so we have inadvertently Define X in under 15 words that's where the Paradox kicks in leaving us in a mind-bending loop to avoid this Paradox in our definition of R's number we must Define it using first order set theory language a simplified version of this can be seen here this definition basically sets up the criteria for Ray's number in the language of first order set theory allowing be's Paradox to be avoided in order to try comprehend how big rayo's number is mathematicians have created the Rayo function which takes in first order set theory symbols to create the biggest number possible from this they have found that the Rayo function remains quite small for values of n up to around 340 and from then on the rate of increase becomes incredibly large all the way up to R's number and so Augustin Rayo won the big number battle with his new unfathomably large number which remains today as one of the largest well- defined numbers ever created
