Disambiguation | ||||
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Would it not be a good idea to have Completeness as a dictionary style definition, leading off to:
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:
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
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)
I agree that if non-mathematical usages are to be included, that should be a separate page. But perhaps in that case, this page should be moved to "Completeness (mathematics)", or, if that's a bit too narrow, "Completeness (mathematical sciences)". Michael Hardy 01:57, 15 Sep 2003 (UTC)
Michael, you said "Not all ordered fields are metric spaces; this is not merely a special case of the metric space case." You are correct, of course, but the concept of a Cauchy sequence is still meaningful in nonmetric ordered fields. Every ordered field has a unique subfield isomorphic to the rational numbers, so a Cauchy sequence is just one in which the difference between every pair of elements for some point onward corresponds to a rational number less than the given real number.
This won't work, (I'm thinking this out, even as I type.), since the differences between the elements in a Cauchy sequence need not be rational. It would work if instead of the subfield isomorphic to the rationals, we used the largest subfield isomorphic to a subfield of the reals. But it's not clear to me that such a subfield must always exist, or that it's always unique.
Please revert if what I wrote was Patent nonsense. But there is a connection between the two definitions, which should be made clear somehow. -- Daran 06:14, 15 Sep 2003 (UTC)
The subection currently reads:
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)
Lambiam, the goal here is to account for the properties of the logical system without regard to meaning or semantics. This is the primary, fundamental, canonical definition of completeness. What you say is true (of semantic completeness), however, that formulation of completeness is a little further down the road from syntactic completeness, so to speak.
Strictly speaking, it is not necessarily true, as you have worded it, that: "A logical system has "completeness" when all true sentences (given the semantics of the logic) are theorems, whereas a formal system has "soundness" when all theorems are true sentences." ...because we can talk about the completeness of logical systems without regard to semantics (the idea that certain sentences are true or false) at all. Pontiff Greg Bard ( talk) 12:30, 27 January 2008 (UTC)
(exdent) May I say that I find the distinction between semantic tautology and syntactic tautology inane? I could likewise define semantic equality and syntactic equality. Take an equation such as 5 + 9 = 2 × 7. Evaluate both sides, using tables of addition and multiplication. If this results in equal numbers, interpreting the entries in the tables as numbers, it is semantic equality. If the same exercise leads to equal strings, interpreting the entries in the tables as strings of digits, it is syntactic equality. Well, the two notions will never give an observable difference, not for these equalities and not for tautologies. The statement that the two always give rise to the same verdict is itself a tautology. We may as well distinguish between normal tautologies, en upside-down tautologies, where the latter are obtained by Australians, New Zealanders, and some yoga practioning logicians. It is somewhat amazing that the author does not distinguish between semantic soundness and syntactic soundness. Whether semantic or syntactic, it doesn't say anything about how to defined the notion for other logical systems than propositional logic. It is hard to imagine a non-semantical method to evaluate quantifications. -- Lambiam 23:39, 29 January 2008 (UTC)
This page is crazy! It is the Ersatz for the deceptively short disambiguation page for "Completion", mixes completely unrelated subjects, such as auditing, autocompletion, and complete metric spaces, etc, etc. I was originally looking for a link to completion of rings in commutative algebra, in order to add it to one of the articles I was editing, and was nary certain it's not even mentioned. Tells you how easy it is to find stuff here! I strongly propose at least to move out the mathematical uses of "complete" and "completion" that have nothing to do with logic, finances, or philosophy into "Complete (mathematics)", as was proposed by Michael Hardy already 5 years ago. Arcfrk ( talk) 08:16, 1 March 2008 (UTC)
What is the difference between maximally complete and syntactically complete as defined? Similarly between deductively complete and semantically complete? (Actually there seems to be a misprint because semantically complete is referred to but not defined.) They seem to say the same thing. The page needs some cleaning up. —Preceding unsigned comment added by 81.210.255.97 ( talk) 17:01, 17 March 2008 (UTC)
In other articles of Wikipedia (eg Predicate Logic) there is reference to syntactic and semantic completeness: the former means that one can always prove P or NOT P; the second means that every statement that holds universally (in every model) can be proved. Are these equivalent? under what assumptions? —Preceding unsigned comment added by 92.50.98.91 ( talk) 09:26, 1 April 2008 (UTC)
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)
L, Do you really believe that it is not generally true that:
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)
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)
I removed (twice now) the following text:
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)
The appropriate thing to do in cases like this is place a citation tag, not delete. That is what those tags are for. It would be nice if people looking for certain terms could find them -- even if certain special people don't use them. Pontiff Greg Bard ( talk) 02:43, 30 January 2009 (UTC)
The formulation is inappropriate anyway. Every sentence being a theorem is a standard definition of an inconsistent deductive system going back at least to Tarski. It has nothing to do with expressing truth or falsehood, and it is a mainstream usage. Even if Gregbard manages to dig a reference for his term, the standard term should come first, and not vice versa. — Emil J. 10:53, 30 January 2009 (UTC)
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)
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)
I removed "and only if" from the three main logical definitions of completeness since this implies soundness is required. If it was indeed "iff", then the correct symbolic version of, say semantic completeness, would actually be: , a formulation which is which is not terribly useful in a discussion of completeness. mjog ( talk) 23:43, 9 January 2013 (UTC)
The comment(s) below were originally left at Talk:Completeness/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
what do you mean, prop calculus is syntactically complete? You just said, for every wff A, either A or its negation is provable. Afaik, neither "p" nor its negation is a theorem of prop calc. |
Last edited at 06:44, 18 April 2008 (UTC). Substituted at 19:53, 1 May 2016 (UTC)
Disambiguation | ||||
|
Would it not be a good idea to have Completeness as a dictionary style definition, leading off to:
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:
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
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)
I agree that if non-mathematical usages are to be included, that should be a separate page. But perhaps in that case, this page should be moved to "Completeness (mathematics)", or, if that's a bit too narrow, "Completeness (mathematical sciences)". Michael Hardy 01:57, 15 Sep 2003 (UTC)
Michael, you said "Not all ordered fields are metric spaces; this is not merely a special case of the metric space case." You are correct, of course, but the concept of a Cauchy sequence is still meaningful in nonmetric ordered fields. Every ordered field has a unique subfield isomorphic to the rational numbers, so a Cauchy sequence is just one in which the difference between every pair of elements for some point onward corresponds to a rational number less than the given real number.
This won't work, (I'm thinking this out, even as I type.), since the differences between the elements in a Cauchy sequence need not be rational. It would work if instead of the subfield isomorphic to the rationals, we used the largest subfield isomorphic to a subfield of the reals. But it's not clear to me that such a subfield must always exist, or that it's always unique.
Please revert if what I wrote was Patent nonsense. But there is a connection between the two definitions, which should be made clear somehow. -- Daran 06:14, 15 Sep 2003 (UTC)
The subection currently reads:
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)
Lambiam, the goal here is to account for the properties of the logical system without regard to meaning or semantics. This is the primary, fundamental, canonical definition of completeness. What you say is true (of semantic completeness), however, that formulation of completeness is a little further down the road from syntactic completeness, so to speak.
Strictly speaking, it is not necessarily true, as you have worded it, that: "A logical system has "completeness" when all true sentences (given the semantics of the logic) are theorems, whereas a formal system has "soundness" when all theorems are true sentences." ...because we can talk about the completeness of logical systems without regard to semantics (the idea that certain sentences are true or false) at all. Pontiff Greg Bard ( talk) 12:30, 27 January 2008 (UTC)
(exdent) May I say that I find the distinction between semantic tautology and syntactic tautology inane? I could likewise define semantic equality and syntactic equality. Take an equation such as 5 + 9 = 2 × 7. Evaluate both sides, using tables of addition and multiplication. If this results in equal numbers, interpreting the entries in the tables as numbers, it is semantic equality. If the same exercise leads to equal strings, interpreting the entries in the tables as strings of digits, it is syntactic equality. Well, the two notions will never give an observable difference, not for these equalities and not for tautologies. The statement that the two always give rise to the same verdict is itself a tautology. We may as well distinguish between normal tautologies, en upside-down tautologies, where the latter are obtained by Australians, New Zealanders, and some yoga practioning logicians. It is somewhat amazing that the author does not distinguish between semantic soundness and syntactic soundness. Whether semantic or syntactic, it doesn't say anything about how to defined the notion for other logical systems than propositional logic. It is hard to imagine a non-semantical method to evaluate quantifications. -- Lambiam 23:39, 29 January 2008 (UTC)
This page is crazy! It is the Ersatz for the deceptively short disambiguation page for "Completion", mixes completely unrelated subjects, such as auditing, autocompletion, and complete metric spaces, etc, etc. I was originally looking for a link to completion of rings in commutative algebra, in order to add it to one of the articles I was editing, and was nary certain it's not even mentioned. Tells you how easy it is to find stuff here! I strongly propose at least to move out the mathematical uses of "complete" and "completion" that have nothing to do with logic, finances, or philosophy into "Complete (mathematics)", as was proposed by Michael Hardy already 5 years ago. Arcfrk ( talk) 08:16, 1 March 2008 (UTC)
What is the difference between maximally complete and syntactically complete as defined? Similarly between deductively complete and semantically complete? (Actually there seems to be a misprint because semantically complete is referred to but not defined.) They seem to say the same thing. The page needs some cleaning up. —Preceding unsigned comment added by 81.210.255.97 ( talk) 17:01, 17 March 2008 (UTC)
In other articles of Wikipedia (eg Predicate Logic) there is reference to syntactic and semantic completeness: the former means that one can always prove P or NOT P; the second means that every statement that holds universally (in every model) can be proved. Are these equivalent? under what assumptions? —Preceding unsigned comment added by 92.50.98.91 ( talk) 09:26, 1 April 2008 (UTC)
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)
L, Do you really believe that it is not generally true that:
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)
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)
I removed (twice now) the following text:
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)
The appropriate thing to do in cases like this is place a citation tag, not delete. That is what those tags are for. It would be nice if people looking for certain terms could find them -- even if certain special people don't use them. Pontiff Greg Bard ( talk) 02:43, 30 January 2009 (UTC)
The formulation is inappropriate anyway. Every sentence being a theorem is a standard definition of an inconsistent deductive system going back at least to Tarski. It has nothing to do with expressing truth or falsehood, and it is a mainstream usage. Even if Gregbard manages to dig a reference for his term, the standard term should come first, and not vice versa. — Emil J. 10:53, 30 January 2009 (UTC)
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)
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)
I removed "and only if" from the three main logical definitions of completeness since this implies soundness is required. If it was indeed "iff", then the correct symbolic version of, say semantic completeness, would actually be: , a formulation which is which is not terribly useful in a discussion of completeness. mjog ( talk) 23:43, 9 January 2013 (UTC)
The comment(s) below were originally left at Talk:Completeness/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
what do you mean, prop calculus is syntactically complete? You just said, for every wff A, either A or its negation is provable. Afaik, neither "p" nor its negation is a theorem of prop calc. |
Last edited at 06:44, 18 April 2008 (UTC). Substituted at 19:53, 1 May 2016 (UTC)