Here is a paragraph on independence from the Wikipedia page (03/12/2015) on Independence in Logic.  It answers your question:

"A sentence σ is independent of a given first-order theory T if T neither proves nor refutes σ; that is, it is impossible to prove σ from T, and it is also impossible to prove from T that σ is false. Sometimes, σ is said (synonymously) to be undecidable from T; this is not the same meaning of "decidability" as in a decision problem."

I agree with Collin's answer above. One can call a sentence which is independent from T, undecidable in (from) T.

A think present in question that I want to shortly address is "arithmetics and algebras that cannot be completed". If we are speaking about theories ( = consistent sets of first order sentences), all theories can be completed, and most countable theories have a set of countable completions [without expanding the language] of the power of continuum. In the classical sense of decidability (existence of a decision algorithm for truth) some of these completions are decidable and most of them are not. This happens alone because of cardinality arguments: there are only countably many decision algorithms. Of course there are also some special cases: the Tarski - Robinson theory Q has only 7 axioms, a set of completions of cardinality of continuum, and NONE of those completions is a complete decidable theory. Something like this is called an essentially undecidable theory. The theory of rings is not so bad. It consists also of finitely many axioms, and has decidable completions (for example the complete theory of the field of complex numbers or the theory of the infinite atom-less boolean ring. and some other) and much more undecidable completions, like the theory of the ring Z of integers.

But of course every sentence independent of the theory of rings is undecidable in the theory of rings (neither the sentence, nor its negation, are provable there).

Of course, Mihai's answer is much more complete than mine! ;)

I guess part of the problem is what "decidable"  means.  For example, in Zermelo-Fraenkel set theory with the axiom of choice (ZFC), the continuum hypothesis CH is often said to be "undecidable."  What that really means is that CH is both consistent with and independent from ZFC .  (I.e., assuming ZFC is consistent, so are ZFC+CH and ZFC+not-CH.)  ZFC is also undecidable in another sense:  there is no algorithmic procedure which, when input codes for the axioms of ZFC (a computable set of numbers) plus a code for a given sentence S in the appropriate first-order language, whether there is a finite proof of S from ZFC.   A theory is complete if, given any sentence in the language of the theory, either that sentence or its negation is a theorem of the theory.  If a complete theory T has a computable list of axioms (finite will do), then you can  start effectively generating theorems from T.  If you're given S, then completeness says that either S or not-S will eventually appear.  Thus T is decidable.  On the other hand, the first-order theory of Boolean algebras was proved to be decidable by Alfred Tarski.  This theory is not complete, as the sentence "there are no atoms" is both consistent with and independent of the theory.    

I think I am finding out something very interesting here.

So, for the case of the Theory of Fields (the theory under The Field Axioms), which is a first order theory, that cannot be completed, would these two statements be identical in meaning and implication?

Sentence S is logically independent.  Sentence S is mathematically undecidable.

In other words: A sentence is logically independent if and only if it is mathematically undecidable.

My reason for asking about independence and undecidability is that I have been under the impression that sentences that are undecidable are always independent, but not the other way round.  If these are the same, I would have greater freedom in my writing to choose the term I prefer.  My guess would be that more readers would be attracted to a title mentioning mathematical undecidability, but maybe the term is more ambiguous and less intuitive than logical independence. My intended readership are physicists, who will not be so well acquainted with either term.

With many thanks,  Steve.

Reference:

https://www.researchgate.net/publication/282647562_Logical_independence_inherent_in_Elementary_Algebra_seen_in_context_of_quantum_randomness

Article Logical independence inherent in Elementary Algebra seen in ...

The difficulty is the multiple use of the word `undecidable'.   Here is some fairly standard notation.   1) A sentence phi is undecidable in  (or independent from) a theory T if both T union {phi} and T union {not phi} are consistent.

2) A theory T is decidable if its logical consequences form  a recursive set.

The easiest way to avoid confusion is to use only independent  from T and not undecidable in the first case.  1) is a property of a theory and sentence. 2) is a property of theory.

Thank you John,

Sounds like the best thing to do.  I think I must stick with the term logical independence.

Thanks to all,  Steve.

I'll go with Paul Bankston's views on independence and ZFC.  However, 'decidable' and 'undecidable' originate in recursion theory. 

An integer function f:N->N is decidable [ie., computable] if there exists a computation which,  given integers j,k,  can decide the truth of  f(j)=k,  in finitely many steps.  The computation must be generated by a single finite program in advance of selecting j,k.  There are uncountably many functions f  and only countably many programs, so most functions are trivially undecidable.  The interesting case is for the countably many definable functions, f0, f1, f2, ...fn,.. .  Integer subscripts n code the function's definition. Every fj  in this discussion is partial -- ie., there may exist k for which fj(k) returns no value.  Total functions, those which always return a value, are a special case of partial.

The interesting case is that not every definable function is decidable.  For example, the function h:N2->N  --roughly defined by h(j,k)=1 if fj(k) returns a value, otherwise =0--  is undecidable.  This is proved from the fact that both the halting function h and the universal function u are definable.  Universal function u:N2->N is roughly defined by u(j,k)=m if fj(k)=m, and u(j,k) is undefined if fj(k) is undefined.  Notice that the halting function h is total in addition to being definable.

I've skipped over semi-decidability.

In my opinion, the logical independency indicates that the logical (and respectively, the conceptual) structures of things/phenomena can be changed without affecting the existing logical (and respectively, the existing conceptual) representations of those things/phenomena. But focusing on mathematical undecidability we realise that 'undecidable' is a very complicated term. In some cases they may be interpreted to be the same and to be identical. In case and only in case there is no logical and mathematical interrelationship, logical dependency, mathematical and logical coherency and semantic intersection between the logical structures of things/phenomena before and after changing, we could be able to conclude that the logical independency and mathematical undecidability are identical. 

@Stefan

Same subject but different question.

Dear Steve,

I very much agree with Baldwin and would use logical independence of a sentence from a theory vs. decidability of a theory. If I may add a note, I would reformulate as follows:

1’) A sentence phi is (logically) decidable in a theory T (w.r.t. proof System S) iff T proves phi or T proves the negation of phi (w.r.t S).

A sentence is (logically) undecidable is the negation, i.e. T neither proves phi nor its negation. If T is consistent, the latter is equivalent to 1) T union {phi} is consistent and T union {non phi} is consistent.

2’) A subset L of words W over an alphabet is (Church-Turing) decidable iff there is a Turing Machine that decides it, i.e. for every w \in L prints 1 and halts if w \in L and prints 0 and halts if w \notin L (equivalently L is recursive)

1’) is a property of a sentence w.r.t. to a theory and a proof system. 2’) is a property of a set of words w.r.t to a model of calculation.

The reason why logical decidability became linked to Church-Turing decidability is it seems to me historic and conceptual (in short: Hilbert wanted an effective method to decide whether a sentence is or is not provable from a theory, implicitly assuming theories were complete, Gödel showed it was not possible within a Hilbert-style proof system and a minimum of arithmetic doing the effective decision and thereby also showed that such theories are incomplete; then Church and Turing further developed recursion theory where decidability became what it now is).

For this reason, I would conceptually rather link the notion

2) A theory T is decidable iff its provable consequences is Church-Turing decidable (recursive),

to the notion of a complete theory.

1’’) A theory is complete iff it logically decides every sentence of its language. (Note an inconsistent theory is trivially complete.)

In 1’’ it’s the theory + the proof system that does the ‘deciding’ of the set of sentences. In 2’ it’s the Turing Machine or the total recursive characteristic function. To get a relation at all, the proof system and the calculation machinery of the Turing Machine need to share some similarities (eg. being able to develop a minimum of arithmetic’s).

However, the history of logic will deceive you here:

One has the following relation (Gödel-Rosser): if a theory T is recursive and consistent and contains a minimum of arithmetic (Peano arithmetic or even weaker Robinson arithmetic) then T is not complete (i.e. it contains logically undecidable sentences).

Another relation is this: Let L \subseteq W and assume T is recursively enumerable and there is a Turing machine which according to some coding calculates the result phi_w for w \in W such that: i) if T proves phi_w then w \in L and ii) if T proves non-phi_w then w \notin L. Then, provided L is Church-Turing undecidable, there is a logically undecidable sentence w.r.t. T.

Logical independence of a sentence from a theory is very widely used (AC or CH is independent of ZF, the axiom of parallels is independent of the other Euclidean axioms). Although this notion is usefull when one does not want to confuse logical decidability of a sentence and Church-Turing decidability (and is widely adopted for exactly this reason), it seems to me that logical independence is rather a semantic, than a syntactic notion. I.e. one could say (but that is a matter of quarrel):

1’’’) phi is logically independent of T iff there is a model of T that satisfies phi and a model of T that satisfies non-phi.

1’’’) and undecidability of a sentence w.r.t. T (as in 1’) are equivalent provided T is consistent and (I guess) the proof system sound and complete.

I am sorry I was so long!

Best,

Eric

Thank you Eric.  Your sumarising statements are very helpful.  I did feel there must be a difference.