I totally agree with Peter.

I think the question I'm trying to ask is:  "if arithmetic is a formal system that is incomplete, for any logically independent proposition I find in that system, can I attribute its independence to the incompleteness -- or are there logically independent statements whose independence is nothing to do with incompleteness?"

I apologise.  I neglected to thank you all for your contributions.  Thank you.

The predicament I am faced with is this.  In a paper I am writing I am following up the work of Toamasz Paterek et al.  I give the reference below.  The authors of that paper use a formal system involving logically independent statements.  The authors say that their system is not incomplete and is therefore not subject to  Gödel's Incompletenss Theorems.  In the paper I am writing that follows their work, I am using an arithmetic that is incomplete and I wonder if I can make any claims regarding incompleteness and the independent statement I use. 

Article Logical independence and quantum randomness

@stefan

So about the best claim I can make would be something along the lines:  I have found an independent statement in a formal system.  That formal system is incomplete and therefore is subject to Gödel's Incompleteness Theorems. ?

The word "incompleteness" has meanings which are slightly different in the situations exposed.

Meaning 1: The axioms of fields do not define a complete theory. (Proof: both R (real numbers) and C (complete numbers)  are fields. Consider the formal statement  S: " exists x such that x2= - 1". R is a model of  "not A" and C is a model of A. Therefore, A is Independent of the deductive closure of the axioms of fields, so this set of sentences (sometimes called also "the theory of fields") is not complete.

Meaning 2: Gödel's Incompleteness Theorem says that any recurrent sets of axioms that are true for the ring of integers Z and is consistent, cannot be complete. [please put the accent on the word "recurrent"] Other said, the theory of Z is undecidable. This means that there is no one algorithm able to decide the truth or falsity (over Z) of all formal statements in the language of Z. (The same can be said for the set of natural numbers R, as Gödel originally did.)

IN CONCLUSION: The word "incompleteness" has very clear meanings in the two situations given in the question, but the two situations are so different, that no connection can be done, as suggested in the question.

HOWEVER: In the book Tarski - Mostowski - Robinson, Undecidable Theories, ways are shown how to interpret Gödel's incompleteness in order to show that some uncomplete sets of statements are also undecidable. Using those techniques they show that the deductive closure of group axioms is undecidable (there is no algorithms permitting us to decide if a formal sentence in the language of groups follows from the group axioms alone) and do the same thing with different theories of rings. I believe that the theoty of fields is in the same situation: not complete and undecidable. Maybe some experts will clarify this point in their posts.

@ Peter

The theory of the structure (Z, +, -, x, 0, 1) - understood as set of all first order statements that are true in this ring - is of course complete and consistent. The point is that this set of statements, seen as set of their codes, is not recurrent and (moreover) even not arithmetical (does not belong to the arithmetical hierarchy). True but unprovable statements can be done only for given recurrent Axiom Systems. They are statements belonging to true arithmetic but not to the deductive closure of the given system. Only some examples are known relatively to Peano Arithmetic: Goodstein's Theorem is one of them. Also, Wilkie and Paris have proven something in this direction.

@ Peter

Fact is, that there are two different ways to define theories (sets of formal sentences). One of them is to give an Axiom System and to refer to its deductive closure. That is the way you understand them (most). There is also another possibility. Take a first order structure (with constnts, operations and relations)  and define its theory as the set of all formal first order statements that are true there. The set you get is always consistent (because there cannot be A such that both A and -A are true in the structure) and complete (because for every A, exactly one of A and -A is true in the structure. Here -A means of course negation). The theory of the ring Z as given above is of course complete and consistent. Is just not recurrent. (that means that is is undecidable if some formal sentence is true over Z. I don't say that you will never prove or disprove that given sentence. I just say that there is no algorithm able to do this decision with all sentences in the language of rings - the decision  if they are true over Z or not.)

Complete theories which have a recurrent axiomatisation are always decidable (this is an easy theorem). This is the way I understand Gödel's Incompleteness Theorem: if a recurrent theory (set of sentences) about arithmetics is sound (correct) and recurrent (has a decision procedure) that it cannot be complete. For example Peano. But if you expand Peano with Goodstein, Ramsey, Hydra and some other new axioms, it remains recurrent and so continues to be uncomplete.

So the moral of the story is that Gödel's Incompletness Theorem is in fact a result about indecidability and not about incompleteness.

Now about theories which are not complete and undecidable, as the theory of all groups. A formal sentence which is true in all groups is of course provable from the group axioms. This is told by Gödel's Completeness Theorem. What is undecidable, is if a given sentence in the language of Groups  can be proved using only the group axioms. So I don't say that there are unprovable sentences true in all groups. If they are true in all groups, they are of course provable. I just say that there is no algorithm able to decide this (given a sentence, to decide if it is true in all groups, or equivalently, if it is provable from the group axioms.) Again: for a concrete sentence, maybe you will prove it somehow from the group axioms, or maybe you find somehow a group where it is false. The point is again, that there are no algorithms able to do this for all sentences.

I hope that this time I was more clear...

Interesting discussion.  Apart from issues of decidability, the term "incompleteness" means the same thing when talking about "Gödel's incompleteness theorem" and when talking about "the theory of fields is incomplete."  The latter is a much easier result to prove:  the existence of an element whose square is the additive inverse of unity is both consistent with and independent of the field axioms in classical logic.  Gödel, through cleverly encoding sentences and proofs of formal arithmetic into arithmetic itself, was able to show that, say, Peano Arithmetic (PA) is incomplete--in exactly the same way the theories of fields, rings, groups, linear orderings, etc., are incomplete.    The Gödel sentence, "I am a sentence true in the structure (N,+,x,0) but not a theorem of PA" is quite artificial, but does the same job as the "existence of the square root of -1" does in the usual first-order theory of fields.  ( In the 1970s, Leo Harrington and Jeff Paris demonstrated a "more natural" Ramsey-theoretic sentence true in N but not provable in PA.)  I might add that the term "completeness" has a second meaning:  Gödel's completeness theorem says that "if a sentence S is true in every model of a first-order theory T,  then there is a formal proof of S with axioms from T."  This sense of "completeness" applies to first-order logic itself.  The term "completeness," as used in Gödel's incompleteness theorem, applies to first-order theories.  I have fond memories of the little joke:  "Gödel's incompleteness theorem disproved his completeness theorem."  Ho ho...

Thanks for all contributions. Coming back to my original question.  Would you guys say that if can construct a self-referential proposition, asserting existence of the square root of minus one, there may be some way of inferring that that existence stems from incompleteness?  Example: Ex i |  i = exp(iπ/2). 

Hi Peter,  It wasn't me who claimed that PA is not a first-order theory; quite the contrary.  It is a non-finitely--but still computably--axiomatized first-order theory.  No doubt about it.  Hi Steve, the whole issue of "self-referential" sentences presupposes a certain amount of expressive power.  The first-order theory of fields--as given in any algebra textbook--comprises a small handful of universal axioms, plus one universal-existential sentence that asserts the existence of multiplicative inverses for nonzero elements.  While I'm not sure of this, I'm confident that there are sound mathematical reasons that "self reference" (whatever that really means) is not possible to express in such a theory. By the way, here's an interesting difference between the theory of fields (F) and Peano Arithmetic (PA):  while both are incomplete first-order theories, F has a computably-axiomatized complete  extension, namely the theory AF of algebraically closed fields.  PA, on the other hand, has no such extension.  (While the theory TA of "true arithmetic," those  sentences true in the standard model (N ,+,x,0), is indeed complete, it is not computably axiomatizable.  This is another consequence of Gödel incompleteness.)

Thank you Ellis.  And where do the real numbers come from?

Hi Akira,

I heartily agree that PA is not (aleph-zero)-categorical because it has countable models with infinite descending chains.  It's not even complete, thanks to Goedel, hence not categorical in any infinite power.  I also heartily agree that there should be more communication between mathematicians and mathematical logicians; here's an example.  When I open a continuum theory textbook, I see the well-known (and beautiful) theorem that an indecomposable metric continuum (compact + connected) has uncountably many pairwise disjoint composants (being the unions of all proper subcontinua containing a given point).  But does "uncountable" mean "aleph-one" or "cardinality of the real line"?  Many topologists don't care and think that is a frivolous question.  But that question is at the heart of Cantor's continuum problem, whose famous solution--I dare say--revolutionized 20th century set theory.  (BTW, the answer is "cardinality of the real line.")