Talk:Decidability (logic)

message: The article says that group theory is undecidable. However this wikipedia article List of first-order theories says that Abelian group theory is decidable. It seems to me that if Abelian group theory is decidable, then certainly ordinary group theory should be. Can someone please clarify?  — Preceding unsigned comment added by 72.48.98.27 (talk) 18:25, 4 January 2013 (UTC)
thread_title:  Inconsistency? 

message: THis is a comment or rather a question.
The article states:
If a system is undecidable, then there exist in it formulas which are not known to be either valid or invalid—perhaps such a formula can be categorized as neither valid nor invalid. Such formulas are said to be "undecided."
Isn't there some confusion with completeness? Decidability means availability of a procedure (algorithm) to figure out the deducibility of any formula without necessarily knowing its demonstration. If a theory is decidable, the algotithm in question can split any set of formulae in deducible (true, valid) and other. The idea of others being either invalid (false?) or indefinite (neither true nor false) comes from another concept.   
Am I right?
81.195.25.24 19:51, 11 September 2005 (UTC)
replies: text: You are completely right, the article was confused. I have fixed it now. -- EJ 19:00, 7 January 2006 (UTC)
thread_title:  CS 

message: "Decidability means availability of a procedure (algorithm) to figure out the deducibility of any formula without necessarily knowing its demonstration. If a theory is decidable, the algotithm in question can split any set of formulae in deducible (true, valid) and other. The idea of others being either invalid (false?) or indefinite (neither true nor false) comes from another concept."
I can fathom from the above why a theory could be decidible but incomplete. You would then have a theory in which the halting problem is solvable (i.e. you could tell that a particular formula either evaluates as true or false, or else causes the checker to never halt, in which case it lands in the "other" pile along with the false formulas). 
:In a first-order theory, and especially in an incomplete one, it does not really make sense to say that a formula "evaluates as true or false". Anyway, to ensure decidability, there must be a checker which always halts, and outputs either "provable" or "not provable". Alternatively, if you insist on separating "true" and "false" sentences, it outputs "provable", "refutable", or "independent". -- EJ 19:12, 20 February 2006 (UTC)
replies: text: There is an example in the article: the theory of true arithmetic is complete, but undecidable. More generally, any complete extension of, say, Robinson arithmetic, is complete and undecidable (Church-Rosser theorem, not to be confused with this one). What you call BMA is an algorithm only as long as you can decide whether something is or isn't a valid proof in the theory. In other words, it only works for recursively axiomatizable theories. -- EJ 19:12, 20 February 2006 (UTC)

text: I forgot to say: re: "Is "decidibility" really about being able to tell whether something is provable independently of its proof?" Yes, indeed, that's the whole point. A decision procedure may in some cases proceed by actually constructing a proof, but it is not a requirement. -- EJ 19:42, 20 February 2006 (UTC)
replies: text: Hey, wait a second, the theory of true arithmetic is not complete, courtesy of Gödel. I would also like a good explanation of how any theory could be complete but undecidable! – Kiewbra (talk) 12:33, 9 January 2013 (UTC)
replies: text: You seem to be confused about the notions involved. A theory is complete, by definition, if for every sentence, it proves the sentence or its negation. The true arithmetic is, by definition, the set of all sentences valid in the standard model of arithmetic \mathbb N. Since the definition of satisfaction in a model guarantees that every sentence or its negation holds in the model, true arithmetic (or the theory of any other model for that matter) is complete. Gödel’s theorem is not applicable, since the theory is not recursively axiomatizable. There are of course much simpler complete undecidable theories: for instance, let L be the language with one unary predicate P and countably many constants cn, n ∈ N, let X be any undecidable subset of N, and let T be the theory axiomatized by the sentences cn ≠ cm for every n ≠ m, P(cn) for every n ∈ N, and ¬P(cn) for every n ∈ N \ X. Then T is complete and undecidable.—Emil J. 13:31, 9 January 2013 (UTC)
thread_title:  Decidablility vs. Completeness 

message: Feferman and Feferman's 2004 biography of Alfred Tarski (p. 193) seems to indicate that this concept was introduced in the 1953 book Undecidable Theories by Tarski, Mostowski, and Raphael Robinson.  This might or might not be worth mentioning in the article. 66.127.52.47 (talk) 20:58, 18 March 2010 (UTC)
thread_title:  essential undecidability 

message: Please someone make really sure that "undecidable" is not confused with "incomplete", especially in this sentence under "Some undecidable theories": "Gödel's incompleteness theorems show that many other sufficiently strong theories of arithmetic [are undecidable]."
Doesn't Gödel show that all sufficently strong theories of arithmetic are incomplete (by constucting an "undecidable"/independent sentence (where undecidable here has another meaning))?  – Kiewbra (talk) 12:29, 9 January 2013 (UTC)
replies: text: Hmm. Undecidability of consistent extensions of Q is usually bundled together with incompleteness theorems, and in fact, it follows easily enough from the Gödel–Rosser incompleteness theorem, since every consistent decidable theory has a decidable complete extension (though the usual proof goes the other way round). However, you are right that strictly speaking, Gödel didn’t prove the undecidability theorem, and even predates the definition of recursive sets. I’m not sure who first formulated the undecidability theorem in this form. The formulation using the concept of essentially undecidable theories comes from was popularized by the Tarski, Mostowski, and Robinson monograph, though Robinson presented the essential undecidability of his arithmetic earlier in 1950, and I suppose the concept of essential undecidability itself is due to Tarski.—Emil J. 13:59, 9 January 2013 (UTC)
thread_title:  Undecidable or incomplete 

message: What is the significance of the "See also" section?  It seems to have names of logicians and years, but I can't match Quine's year to a specific event or paper. — Arthur Rubin  (talk) 16:02, 21 January 2014 (UTC)
thread_title:  See also 

message: I'm no illiterate on the subject, but aren't the Decision problem and Undecidable problem kinda dealing with the same stuff as this one? --uKER (talk) 22:52, 19 July 2018 (UTC)
thread_title:  Redundant articles maybe?