Like most, I'm having a hard time understanding the consequences of Gödel's Incompleteness Theorems.
In particular, I'd like to understand their connection to the concept of infinite mathematical structures.
In doing so, I hope to formulate a better opinion on the merits of constructivism and finitism in regards to Gödel's theorems.
Without being philosophical, I want to know whether a given formal system built from constructionist principals (finite mathematical objects), would be complete, and whether Gödel's arguments say anything about these kinds of systems.
Taken together, the two theorems can be informally stated as followed:
First incompleteness theorem (Godel-Rosser): Any consistent formal system S within which a certain amount of elementary arithmetic can be carried out is incomplete with regard to statements of elementary arithmetic: there are such statements which can neither be proved, nor disproved in S.
Second incompleteness theorem (Godel): For any consistent formal system S within which a certain amount of elementary arithmetic can be carried out, the consistency of S cannot be proved in S itself.
(I'm quoting from a book called Gödel's Theorem: An Incomplete Guide to Its Use and Abuse)
In both of these cases they say "a certain amount of elementary arithmetic".
But what does that mean in regards to "infinity"? Does that mean a requirement for incompleteness is an infinite amount of objects capable of arithmetic (i.e. like an infinite amount of numbers (i.e. the natural numbers))?
Or maybe stated in terms of Peano arithmetic: "For every natural number n, S(n) is a natural number."
There is incompleteness in the arithmetic of this system because you can always call a successor function to get another number?
These are the specific questions I have around the subject:
1. If a system has a finite amount of numbers for arithmetic, can the system be complete?
2. If ZFC does not have the axiom of infinity, can the system be complete?
I have an infinitesimal amount of experience in mathematics, so I appreciate your indulgence.