Consistency of Axioms

Question

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 A_i such that all possible combinations are generated. So, we get theorems derived from individual Axioms and :

1. Set of all theorems derived using axioms : A_i & A_j such that i != j; i,j belong to {1,2,3,4}
2. A_i & A_j & A_k such that i != j != k ; i,j,k belong to {1,2,3,4}
3. 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 A_is, pairs- A_i with every A_j, triplets- A_i with A_j with A_k (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.

Answer 1

The basic confusion in the submitter's argument is a misunderstanding of what is meant in the quote by 'the next theorem'. One must distinguish between those theorems which may be proved in principle, which is what the submitter's argument refers to, and those theorems which have actually been proven (and physically written down!) in fact, which is what the quote refers to. It is a brute physical fact that we limited beings can never in fact prove more than a finite number of theorems in any system, even though the system most likely has infinitely many theorems which we in principle could prove. Therefore, if a system is inconsistent, there must be some contradiction which in principle may be derived within the system - but that doesn't mean that we have in fact already found that derivation yet! This is what the quote is saying.

Answer 2

It is possible to have inconsistent axioms, say, A1 and A2, such that for example A1 → C and A2 → ¬C, and yet such that they nonetheless both imply the same B, i.e. A1 → B and A2 → B.

You will be able to prove B but not C from the set of axioms {A1, A2}. C will in fact show that the axioms A1 and A2 are inconsistent.

Thus, proving some B doesn't prove there is no C for which the axioms are inconsistent.

It is easy to construct a simple instance of such axioms:

A1 : B ∧ C

A2 : B ∧ ¬C

Here, both A1 and A2 imply B, but they are inconsistent as to C: A1 implies C while A2 implies ¬C.

Answer 3

The OP writes:

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.

Consider the following axioms:

1. A & ~A
2. B
3. C

From these axioms I can derive the following theorems:

• A & B from axioms 1 and 2
• B & C from axioms 2 and 3

Those theorems do not contradict each other, but as soon as I derive ~A from axiom 1, I would have a theorem that contradicted the first theorem above.

If I check that the axioms are consistent then I can trust that the theorems would be consistent as well, but I have to first check that those axioms are consistent. One could try to check that all the theorems are consistent but that involves many more statements than just checking the axioms. It would be simpler to just check the axioms.

The OP also writes:

It only makes sense to worry if and only if my current theorems cannot fully express the exhaustive essence of my axioms.

From any finite set of axioms there are an unlimited number of statements that could be derived from them. For example, from the first axiom I could derive A, then A & A, then A & A & A, and I could go on indefinitely. Or, I could derive A, then ~~A, then ~~~~A and so on. These are not interesting, but they are all derivations from axiom 1.

Here is the question:

So, the question is, why should I worry about new theorems, and, why any one theorem cannot fully express an axiom(s)?

Infinitely many statements can be derived from any finite set of axioms. The point of having a small set of axioms is so they can represent all of these statements. It is to focus on that finite set of axioms, checking just them for consistency rather than checking all of the statements that we might happen to derive from them.

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Nagel, E. and Newman, J. R. Godel's Proof. 1958. The OP references this page: https://archive.org/details/gdelsproof00nage/page/14