Talk:Completeness

message: Would it not be a good idea to have Completeness as a dictionary style definition, leading off to:
* Completeness (metric spaces)
* Completeness (measures)
* Completeness (logic)
etc?
-- Anon
Yes, I agree that this would make a good disambiguation page.
However at this point, it's not really necessary; the metric (or uniform) space concept is the only one that has had enough written about it to form an entire article.
Thus, I have created Complete_space, which currently redirects here, in anticipation of this, and we can link to that page instead of this if we wish.
However, I don't think that it's necessary to move the contents of this article over there, or to move links to this article over there, until we reach the point that there is either:
* a great deal written about completeness in some other article (such as the article on lattices) that justifies branching it off to a new article, or
* a great deal written about one of the other sorts of completeness in this article that justifies splitting this into two articles.
Then we can do the transition; otherwise, the move may be harmless but is largely a waste of time.
IMO.
-- Toby Bartels, Monday, June 10, 2002
The main content of this page may need to be moved to complete_metric, or something similar, but it would not be sensible to do this until Pages_that_link_here is fixed.
-- Anon
If and when we move it, it should be to Complete_space, since we'll want to discuss other notions of complete spaces (complete uniform spaces, maybe even complete Cauchy spaces) under the heading Generalisations. — Toby Bartels, Sunday, July 14, 2002
Given that Complete measure is a separate article, I'm doing the move now.
-- Toby 23:40 Feb 20, 2003 (UTC)
-----
I removed
:If V is separable, it follows that any vector in V can be written as a (possibly infinite) linear combination of vectors from S.
Linear combinations are by definition finite, so we would have to say a bit more to make this statement precise. Even then, I'm not sure that it is true in general (it surely works in Hilbert spaces though). AxelBoldt 18:12 29 May 2003 (UTC)
Our article now discusses infinite linear combinations.
-- Toby Bartels 09:43, 26 Aug 2003 (UTC)
thread_title:  completeness 

message: The subection currently reads:
:In proof theory and related fields of mathematical logic, a formal calculus is said to be complete with respect to a certain logic (i.e. with respect to its semantics), if every statement P that follows semantically from a set of premises G can be derived syntactically from these premisses within the calculus. Formally,  implies  . Especially, all tautologies of the logic can be proven. Even when working with classical logic, this is not equivalent to the notion of completeness introduced above (both a statement and its negation might not be tautologies with respect to the logic). The reverse implication is called soundness.
Two things:
(i)This notion is not really native to proof theory, although it originated there, but is rather one of the fundamental ideas of model theory.
(ii)Completeness is not restricted to logic; it makes perfect sense to talk about completeness wrt to any first-order theory. ---- Charles Stewart 11:52, 29 Sep 2004 (UTC)
thread_title: Proof theory subsection

message: Below follows a list of articles about some form of completion or completeness. Note that I did not check all of them to see whether it is appropriate to treat them here.  --Lambiam 16:51, 20 March 2008 (UTC)
thread_title:  Please disambiguate this 

message: L, Do you really believe that it is not generally true that:
:"In mathematics, the notion of completeness is related to completeness in logic."
It seems that there is a tendency to write-out everything that connects math to logic. This is the math-centric thing I am always talking about. 
It's another small point, but over time the consequences of this tendency have accumulated. Pontiff Greg Bard (talk) 22:59, 28 March 2008 (UTC)
replies: text: To the extent that everything in mathematics is related to logic, it is true that the notion of (say) generalized function in mathematics is related to logic. However, that does not imply that the notion of generalized function in mathematics is related to generalized function in logic. In general, it is a nonsequitur to conclude from the fact that field A is related to field B, to the relationship of equally named concepts in these fields. The various things that mathematicians have called "complete", "completion" or "completeness" have hardly any relationship to each other beyond the common human-language meaning of complete. How does completeness of a measure relate to completeness of a formal system? About to the same extent as it relates to completeness of a dinner set.  --Lambiam 01:13, 29 March 2008 (UTC)
thread_title:  Math v logic again 

message: The diode logic article mentions that "diode-resistor logic ... is not a complete logic family."
Is there already an article that discusses the various kinds of "complete logic family", and what makes them complete?
Perhaps under some other name?
Or should I start such an article?
--68.0.124.33 (talk) 04:32, 2 October 2008 (UTC)
thread_title:  yet another kind of completeness 

message: I removed (twice now) the following text:
:A formal system is extremely complete or complete in the extreme sense iff every sentence is a theorem. Not all formal systems are intended to express "truths" or "falsehoods," however for those that do an extremely complete formal system is the same as an inconsistent formal system.
Until or unless a citation for this particular meaning of 'extreme completeness' is forthcoming, this text doesn't belong in the article. Zero sharp (talk) 01:33, 30 January 2009 (UTC)
replies: text:  I eagerly await your citation, which I'm sure you just didn't have right at hand when you were inserting your 'special' usage of the term. Zero sharp (talk) 07:51, 30 January 2009 (UTC)

text: Quelle surprise, no citation, no hits on google for anything other than this page or mirrors of it for the phrase 'extremely complete', therefore removing (for the third time now) as unsourced neologism. Zero sharp (talk) 22:32, 12 February 2009 (UTC)
thread_title:  "Extreme Completeness" 

message: I have my problems with the paragraph
If V is separable, it follows that any vector in V can be written as a (possibly infinite) linear combination of vectors from S.
I doubt that the statement is correct. Take for example the prominent separable Banach space V:=C[0,1], the space of continuous real valued functions on the unit interval [0,1] with the topology of uniform convergence and take S:={1,X,X^2, ...} to be set of all monomials. Then, by Weierstrass Approximation Theorem, S is a complete subset of V. However, an infinite linear combination of monomials is a power series. And a power series which converges always gives you a differentiable function (even a real analytic function). But there are functions in C[0,1] which are not differentiable. So, the statement must be false. Unless I dont understant the statement correctly. I hope, someone can explain this.--131.234.106.197 (talk) 17:05, 23 February 2011 (UTC)
thread_title:  Functional Analysis 

message: I think the current definition of inconsistency is misleading in one case and wrong in others: "A formal system is inconsistent if and only if every sentence is a theorem."
This equivalence holds in some systems, e.g., classical first-order logic, but not in others, e.g., paraconsistent logic. Even in the systems where it holds, in all of the dozens of books that I have read, this fact is proven as a theorem and even has special names, e.g., the inconsistency effect or principle of explosion. I think the definition should be changed to agree with usual practice: a theory or formal system is inconsistent iff some formula and its negation are both theorems. Also, on the stricter logical meaning of "sentence" (a formula with no free variables), the equivalence as stated above is not as strong as it could be because it holds for all formulas. I will gladly change the definition if no one has objections. Cheers, Honestrosewater (talk) 05:06, 15 January 2012 (UTC)
replies: text: I'm not even sure a definition of inconsistency is needed here since a) an article about completeness and b) where consistency it is used in the definitions of completeness, it is linked to the page on consistency. I'll remove it. mjog (talk)  —Preceding undated comment added 23:28, 9 January 2013 (UTC)
thread_title:  Definition of an inconsistent system 

message: Substituted at 19:53, 1 May 2016 (UTC)
thread_title: Assessment comment