User:Googology Noob/FOST golf

We have a page for Turing machine golf, so why not FOST golf? This page assumes familiarity with FOST and how it works. A good page for a basic guide is this.

This page will contain a group of challenges to construct a certain set in the language of FOST. Answers are preferably in pure FOST using only ∃ ∧ ¬ ∈ =, though feel free to post a sketch of an answer in pseudo-FOST. Adding challenges is encouraged, but try to add them approximately in order of difficulty.

If a question asks to construct a certain number/ordinal then unless stated otherwise, it is expected to construct the corresponding Von Neumann ordinal. =Challenges= I will divide this into a few parts: functions, numbers/arbitrary sets and ordinals.

Addition
Define addition. That is, construct the set of all pairs {{a,b},c} such that a+b=c for natural numbers.

Multiplication
Define multiplication. That is, construct the set of all pairs {{a,b},c} such that a*b=c for natural numbers.

Exponentiation
Define exponentiation. That is, construct the set of all pairs {{{a,b},b},c} such that a^b=c for natural numbers

Ackermann function
Define the Ackermann function.

Division
Define division.

FGH
Define the FGH for any natural number and for any ordinal at least up to \(f_{\omega^2}(n)\).

Counting elements
Define a function which counts how many elements a given set has.

BEAF
Define BEAF at least up to linear arrays of arbitrary length.

BB(n)
Define BB(n), the busy beaver function.

Thousand
Define the number thousand.

Googolplex
Define the number googolplex.

Primes
Define the set of all prime numbers.

Graham's number
Define Graham's number.

Coprimes
Define the set of all pairs {a,b} such that a and b are coprime (that is, their greatest common divisor is 1).

\(\Sigma(1000)\)
Define BB(1000).

\(\omega\)
Define the set of all natural numbers, \(\omega\).

\(\omega^2\)
Define the set of all ordinals smaller than \(\omega^2\).

Addition
Define ordinal addition.

Multiplication
Define ordinal multiplication.

\(\omega^\omega\)
Define \(\omega^\omega\).

\(\varepsilon_0\)
Define \(\varepsilon_0\).

Exponentiation
Define ordinal exponentiation.

Ord
Define the set of all ordinals.

Binary phi function
Define the binary phi function for all ordinals.

\(\Gamma_0\)
Define \(\Gamma_0\).

\(\vartheta(\Omega^\omega)\)
Define the SVO.