Gödel' Incompleteness Theorem (GIT) states that certain theories $T$ (formal systems that are of sufficient complexity to express the basic arithmetic of the natural numbers and which are consistent, and effectively axiomatized) e.g. Peano Arithmetic, are incomplete, i.e. there are statements $S_k$ that can't be proven right or wrong within the theory, by which I mean composing the axioms of the theory.
The way out is to accept the unprovable statement $S_1$ either as true or false, which by GIT leads to another theory that is again incomplete. In fact even two theories! Let's choose "false" for the moment and I'll write $\lnot S_1$ for that. If we go on (turning coffee into theorems), we'll reach the next unprovable statement $S_2$ (which might depend on $S_1$ being false) and are left with another decision, whether to accept this as true or false. Now we already have five(!) theories:
- $T_0$
- $T_0\cup\{S_1\}$
- $T_0\cup\{\lnot S_1)\}$
- $T_0\cup\{\lnot S_1,S_2\}$
- $T_0\cup\{\lnot S_1,\lnot S_2\}$
This approach can be continued. Our resulting (incomplete) theory is described by a sequence of unprovable, but as true or false accepted, statements. Properly arranged this looks like a tree graph.
What is known about this tree? Does it have loops? (No loops, see Alex' comments)
I'm also looking for references...