User blog comment:P進大好きbot/List of Common Failures in Googology/@comment-32697988-20190706142929/@comment-35470197-20190706150944

1. What do you mean by "term" and "recursive system"?

A term traditionally means a constant or a variable. If you are working in Peano arithmetic, a term is a natural number. If you are working in \(\textrm{ZFC}\) set thoery, a term is a set.

A recursive system means a tuple consisting of recursively defined language, recursively defined functions (i.e. computable functions), recursively defined fundamental sequences, recursively defined orderings, and so on.

2. I can't understand the problem well. Is the problem \((,(,))\) having multiple divisions \((\#_1,\#_2 )=("","")\) and \((\#_1,\#_2)=(",(","),")\)?

Right. Why do you think that you could not understand the problem? You actually pointed out the problem.

3. What do you mean by "overloaded equation"?

It means a relation written by the equation symbol which is not the equality of terms originally defined in the base theory. For example, if you are working in a first order logic such as Peano arithmetic or \(\textrm{ZFC}\) set theory, then the equality is assumed to satisfies the logical axiom. On the other hand, if you define a relation written by the same symbol, it is not required to satisfy the logical axiom.

4. Isn't it when such a googologist tries to imitate oracle busy beaver function instead of regular busy beaver function?

No. I intended the original busy beaver function, which actually uses oracle. Of course, oracle busy beaver function might be another one of stuffs which such a googologists imitate, too.

5. What do you mean by "the definability of a natural number in the base theory by an L-formula"?

It means the predicate which judges whether a given natural number \(n\) is characterised by \(F(n) \land \forall x,(F(x) \to (x=n))\) for an \(L\)-formula \(F(x)\).

6. What is "property"?

it means a formula. A formula \(F(x)\) defined a term \(t\) if \(F(t) \land \forall x,(F(x) \to (x=t))\) holds.

7. Do you mean "expression"?

Right.