In Godel's Proof by Nagel & Newmann, they write :
In Riemannian geometry, for example, Euclid's parallel postulate is replaced by the assumption that through a given point outside a line no parallel to it can be drawn. Now suppose the question: Is the Riemannian set of postulates consistent? The postulates are apparently not true of the space of ordinary experience. How, then, is their consistency to be shown? How can one prove they will not lead to contradictory theorems? Obviously the question is not settled by the fact that the theorems already deduced do not contradict each other—for the possibility remains that the very next theorem to be deduced may upset the apple cart.
I think that if current theorems do not contradict each other, it does imply that the very next set of theorems derived will not contradict each other. Let there be Axioms A1, A2, A3, A4. Then let us apply logical operations (rules of manipulation) on each Ai such that all possible combinations are generated. So, we get theorems derived from individual Axioms and :
- Set of all theorems derived using axioms : Ai & Aj such that i != j; i,j belong to {1,2,3,4}
- Ai & Aj & Ak such that i != j != k ; i,j,k belong to {1,2,3,4}
- And a theorem derived using all four axioms: A1, A2, A3 & A4
Let us also suppose that none of these obtained theorems contradict each other. Now only logical operations (manipulation of symbols) can be applied to these theorems, or in fact to derive any new theorem. But the essence of each axiom has been captured in the set of theorems I have constructed above. If they could not contradict each other, why should I worry about any new theorem contradicting another? It only makes sense to worry if and only if my current theorems cannot fully express the exhaustive essence of my axioms. So, the question is, why should I worry about new theorems, and, why any one theorem cannot fully express an axiom(s)?
EDIT:
The main idea I am advocating is the following:
If there are four axioms, it must be sufficient to have one instance of every type of combination i.e. singulars -all individual Ais, pairs- Ai with every Aj, triplets- Ai with Aj with Ak (triplets) and quad- any one theorem which employs all four axioms. The idea is to capture all cross interactions. That is, in my argument, [ A1 & A1 & A2 & A2 ] should not offer any extra insight than theorem [A1 & A2] since what is important is the cross interaction of axiom A1 with A2.