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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 :

  1. Set of all theorems derived using axioms : Ai & Aj such that i != j; i,j belong to {1,2,3,4}
  2. Ai & Aj & Ak 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 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.

  • Maybe I’m missing something. From what you quote, it looks like the authors are talking about a system where we swap one axiom for an alternative “competitor”. But your argument seems to discuss adding new Theorems (which must be mutually consistent with the system before, by the nature of theoremhood). Does this relate to their point? – Paul Ross Jul 30 at 11:56
  • @PaulRoss I'm just focussing on their argument that it is indeed possible for next theorem to be inconsistent. What I find difficult to understand is why? If I have deduced initial theorems using all axioms in all combinations, each axiom has been captured in all combinations. If this set has no contradiction, any other set derived henceforth should not have any contradiction. But since they claim that it can be the case, I am interested in knowing why? Why would a contradiction not show at (say) 'inital' stage, but at 'later' stage? The essence of axioms must fully reflect at every stage. – Ajax Jul 30 at 13:48
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    No, that is not true. If A1 and A2 are together inconsistent that does not mean that everything that follows is inconsistent. The axioms P→Q, Q→R, P&~R are inconsistent. You can derive from them P→R. But that itself is not inconsistent. – Eliran Jul 30 at 19:10
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    @NickR What I intend to say is that because inconsistency lies, ultimately in 'togetherness' of two or more axioms, the idea of inconsistency must also reside in any theorem formed using two (or more) inconsistent theorems. An application of a logical operation cannot, by itself, impart inconsistency. Therefore, inconsistency must reside beforehand. Therefore, what I want to ask is why was inconsistency discovered later (after application of multiple operations/theorems), and not earlier (as soon as I combined inconsistent axioms in any manner i.e. in any theorem formed). – Ajax Jul 30 at 20:48
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    Ajax, I think your point is basically right, in that if the axiom set as a whole is inconsistent then it was always inconsistent, even if we didn't know that yet at the time. But the authors are asking the epistemological question about investigating new axiom systems and asking whether we can prove we aren't just wasting our time on a body of work that will ultimately be invalidated by an inconsistency proof. If you keep reading the book, you'll find out more about why a formal approach doesn't quite work in giving us the confidence we're looking for. – Paul Ross Jul 31 at 6:44
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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.

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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.

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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.


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

  • So does this effectively means that theorems are formal statements, and axioms are more like 'ideas'? That theorem can never fully capture entire idea, but only in parts. Only this would explain why consistent theorem don't imply consistent axioms, but consistent axioms imply infinite consistent (derivable) theorems. – Ajax Jul 30 at 14:14
  • AND If that is the case, then we should not be able to formalise axioms, for ideas cannot be fully captured (written in a statement). If a statement can fully capture an idea, then I can write a statement (or theorem), and compare two such axiom theorems for inconsistency, and because by construction they fully reflect axioms -therefore inconsistency must be visible. – Ajax Jul 30 at 14:15
  • Plus, see edit. It makes my point clearer. – Ajax Jul 31 at 7:57
  • @Ajax In general ideas cannot be fully captured in written statements. There is also "tacit knowledge". However, that's a different topic. See Michael Polanyi's Personal Knowledge. The axioms can be looked at as a subset of the theorems. There may be different sets of axioms. If we can show this subset is consistent then all the theorems that can be derived from those consistent axioms are consistent. It is easier to show that a subset is consistent than to show all the theorems are consistent. Even with all the interactions we don't know if a contradiction isn't within a theorem or axiom. – Frank Hubeny Jul 31 at 11:20

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