Uncomputable functions

An uncomputable function, is a function that cannot be calculated by a Turing machine, and therefore cannot be calculated by any possible computational system. Uncomputable functions can be used in googology to make very big numbers.

Busy beaver function
The busy beaver function is a notable example for explaining how a function can be uncomputable, before understanding what the BB function is, you must know what a 2 color Turing machine is. The busy beaver function is a function that outputs the biggest possible output a 2 color Turing machine can give, with an input number of states. A more specific definition is that BB(n) gives an output of the largest finite number of ones that can be outputed by a turing machine that only uses ones and zeros, and has n states. The reason that this function is uncomputable is that a turing machine cannot calculate it, because if we assume a turing machine could calculated it, such an output of a turing machine would be included as one of the outputs of the BB function, therefore making a contradiction, since a function can`t output a number as big as itself. Since this function will eventually give an output bigger than any output a turing machine could give, it is faster than all computable functions, including but definitely not limited to, the g sequence, friedman`s TREE sequence, Buchholz Hydra function, and D(D(D(D(D(n)))). It is also possible to define BB functions with more than just 2 symbols, these functions grow faster than the regular busy beaver numbers, but are not significantly faster. Another variant of the BB functions is the higher order BB functions, these functions are the same as BB, but use oracle turing machines instead of regular turing machines, oracle turing machines have a halting oracle, which solves the halting problem for lower order turing machines.

Maximum shifts function
The maximum shifts function is very similar to the busy beaver function, but instead of outputting the largest number of ones for a turing machine of n number of states, it outputs the largest number of state transitions for a turing machine of n states. Since printing BB(n) ones on a turing machine tape requires at least BB(n) steps, S(n) is at least as fast, if not, faster than BB(n).

Xi function
The Ξ function is another function which gives you the biggest output for a certain number of symbols in SKIΩ calculus, which is turing complete, so it is also uncomputable and faster than all computable functions. Here is how SKIΩ calculus works:

Ix → x

Kxy → x

Sxyz → xz(yz)

if x reduces to I : Ωxyz → y

else: Ωxyz → z

A SKIΩ calculus program halts when none of the conditions above apply (for example: Kx). Because of the Ω operation, Ξ(n) is actually faster than BB(n).

Rayo`s function


rayo(n) outputs the smallest positive integer bigger than any finite positive integer named by an expression in the language of first-order set theory with n symbols or less. The reason this function is so huge is because you can define the BB function, the maximum shifts function, and the higher order BB functions, in first order set theory. This means that rayo(n) eventually is faster than these functions. The language of FOST that Agustin Rayo defined has the following operations: The variables in FOST can be any set possible, but the ∃a(e) narrows down what possible sets these variables can be, for example, to make an empty set in FOST you need to use the following expression: (¬∃x2(x2∈x1)), this could be translated to English as "don`t make the second variable allow for the formula "the second variable is an element of the first variable" to be true" or as "make it so that the second variable cannot contain the first variable". Once the creator of the Xi function (Adam Goucher) claim his function was faster that rayo`s number, but this was proven false, since Adam Goucher misunderstood what Agustin Rayo define as First Order Set Theory, as, First Order Arithmetic (which is weaker than FOST).

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