760:. A good way to understand these things is like moving boxes. If you have a map (or a checkerboard or whatever), and some rules you can move boxes to some places and not others. The places the rules allow you to move a box is a theorem. Perhaps more appropriate to the analogy, the box is a theorem in some places not others. So this is a totally new understanding of theorem. It doesn't involve "true" or "proved." Calling a box a theorem will certainly seem strange to a mathematician, however this is what is going on behind the scenes. It is not appropriate to say "Well but it isn't REALLY a theorem." That would be missing the point entirely. It is exactly this that is the essence of theorem. This is what it is like to talk about syntax without semantics. The same analogy goes for tautologies, although I found theorem easier to explain.
1369:
equivalent under mild assumptions that would tend to hold in logics unless they are specifically constructed to form a counterexample. Following the definitions of the article, maximal completeness implies syntactic completeness, but the converse implication does not necessarily hold. Take the logic whose sentences consist of a finite sequence of ¬ symbols followed by the symbol ⊥. The theorems are those sentences that have an odd number of ¬ symbols: {¬⊥, ¬¬¬⊥, ¬¬¬¬¬⊥, ...}. This logic is clearly syntactically complete. But ⊥ is not a theorem, and this sentence is also not a negation and therefore not the negation of any theorem. So the logic is not maximally complete. In general, a necessary condition for maximal completeness is that all unprovable sentences have the form of a negation; you can't have
1095:
We are able to move symbols around on paper according to rules, and get other symbols that happen to preserve deductive qualities. This is the level of analysis we seek, because we want to make sure we have a legitimate deductive system, with no presuppositions (as you have made about the so-called "equal numbers" you speak of.) We can see today that Euclid fudged somewhat on the rigor of some of his proofs. I'm sure he thought it was okay at the time. Mathematics is orders of magnitude more complex today, so BE CAREFUL! ...and be well.
2043:. 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,
22:
2012:
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 which are not differentiable. So, the statement must be false. Unless I dont understant the statement correctly. I hope, someone can explain this.--
794:
fine. However, it should not be looked upon as foreign or hostile at all. Please look at it as filling in the philosophical perspective. It seems to me that in philosophy classes we are told that mathematicians study this area too, but the mathematicians are not told that we study this area too (or they are told and promptly shrug them off as pretenders --silly).
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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.
1071:, 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
723:
understand what you say you understand, and you are forthright on these points where you are not 100% clear. However, I think what you do understand is not permitting this new understanding. This actually is the more fundamental understanding, not trivial at all, perhaps you have been mathematically prejudiced.
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certain symbols as constants. Otherwise, suppose we have a logic with 0-ary symbols ⊥ and ⊤. We can interpret ⊥ and ⊤ as, respectively, "false" and "true" in some models, but as "true" and "false" in other models. That means that basically no sentence holds in all models. If the logic is semantically
1094:
You understand that this kind of supports my whole "mathematical prejudice" theory do you not? You are going to interpret a number as a number, huh? Yeah that seems pretty silly. Only that is not what is going on here. It's not a number until it is interpreted as such, until then it is just a symbol.
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There are no deadlines on wp, so I don't really have a good answer. The next time I get to the university library perhaps? It seems that the article is tagged for no sources so it really boils down to what we agree on until there are. I guess I could rightly ask you when you will come up with sources
315:
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
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If by FS you mean the system with theorems of the form †∗†∗ then I must confess I haven't the foggiest idea what it means for a string of daggers and asterisks to be a tautology, and so your statement that this system may qualify as not being complete in your sense of the term carries no content for
1207:
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
1079:
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
722:
Well
Lambiam, let me first say that I appreciate your care and attention to these matters. Our discussion has been a pleasure on my part. As you may know, I am not always very diplomatic, so please forgive my forthright conjecture. I can tell you have studied this subject well. It is clear that you
160:, 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:
1968:
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
1345:
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
1289:
I agree with the disambiguation proposal to merge. I would prefer it if we merely designated this article space as the disambiguation page without the "(disambiguation)" in the title. I'm not so keen on separating out the "(mathematics)" however. I find that whole (very strong apparently) tendency
987:
A semantic tautology is a wff of truth-functional propositional logic whose truth table column contains nothing but T's when these are interpreted as the truth-value Truth. A syntactic tautology is a wff of truth-functional logic whose truth table column contains nothing but T's when these T's are
2011:
I doubt that the statement is correct. Take for example the prominent separable Banach space V:=C, the space of continuous real valued functions on the unit interval with the topology of uniform convergence and take S:={1,X,X^2, ...} to be set of all monomials. Then, by
Weierstrass Approximation
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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
557:
I think your statement "...do you just mean that "A = A" is not true because "A" is a variable?" is another way of looking at the idea I am trying to explain. These are syntactic qualities. I will look further into these matters and try to get a good example. In lieu of that good example, I will
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right? I'm going to get back to you at some point for your example. It is the formation of the symbols, "→", "p", etc, that determine whether it is a tautology or not, not any interpretation of, for instance, p being true. So for instance, if your example is a tautology, then it is a token or the
290:
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
286:
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
1441:
If the logic is sound and syntactically complete, then it is also semantically complete: every true sentence is a theorem. The argument is as follows. Let P be an arbitrary true sentence. Then ¬P cannot be a theorem, since the system is sound. But then instead P is a theorem, since the system is
793:
I hope at some point this results in an "aha moment" rather than impatience with myself. I am only a cab driver, so I am learning a lot more from others than I am contributing. However, these appear to be concepts that mathematicians in general do not care about or, pay attention to, and that is
630:
I guess my next question is what does "carries no content for me" mean? Does it mean a) you just would like more clarification, b) you are indifferent to the clarifications I have made, and will be going ahead with your version, or c) you understand it as trivial, or d) something else? The other
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page is now a redirect to its more common meaning, related to drilling wells - which has nothing to do with the concept of mathematical completeness. Every page that links to that article uses it in that context. This talk-page is the only place that links to "well completion" to refer to the
1368:
I'm not familiar with the terminology of "semantical completeness", but I suspect the property may be meant that is called "strong completeness" in the section on
Mathematics. I'm also not sure how common and notable all these variants of the notion of completeness in logic are, since most are
1398:
In other articles of
Knowledge (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
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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. ----
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It carries no content for me in the sense that I am unable to assign meaning to that string of words, presumably in the same way you can't extract meaningful content from a statement that "the rights of cloud precede mutation". In technical terms, the utterance is inoperative as a
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for your language too, however I'm not that way. I am quite sure there are sources that support both formulations, however yours belongs a little later in the article because of it involves an interpretation of the pure thing we are talking about. There is no good statement about
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the sentences ⊥ and ⊤, and nothing else. The logic has a single axiom, ⊤, and no derivation rules. This logic cannot be syntactically complete, since neither ⊥ nor ¬⊥ is a theorem – the latter is not even a sentence of the language! But the system is semantically sound and
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uninterpreted tokens rather than, say, truth-values. The rules for generating the truth table column tell us to use one of these uninterpreted T's in exactly those cases where semantic considerations would have led us to use the truth-value Truth. This can be found at this
2038:
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
1879:
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?
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number of ¬ symbols: {⊥, ¬¬⊥, ¬¬¬¬⊥, ...}. This logic is clearly syntactically complete. The true sentences are those with an odd number of ¬ symbols, but none of these is a theorem, so this logic is semantically incomplete. It is also very unsound.
277:
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)".
631:
question I have is how long are you willing to wait for (1) and (2)? I think the fact that this type of formula can be a non-tautologous theorem should really inform and enhance your notion of tautology. That's is what I would like for the readers.
1260:; it is zany that the latter use of the term, which is not even mentioned in any dictionary I have access to and does not occur (except for the Knowledge article) in the first 100 Google hits for "completion", should have primacy. Funny fact:
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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.
1933:
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.
1275:
now. However, I consider mathematical logic as much a field of mathematics as category theory, graph theory, analysis, and algebra, and I strongly disagree with the idea of banning the meanings in mathematical logic from such a page.
346:, 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.
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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.
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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:
1149:
here. The distinction between the two notions of tautology is, however, really mainly silly because they necessarily always coincide. Would you ask someone who states "¬(A ∧ ¬A) is a tautology" to clarify whether they mean a
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Without the requirement of soundness, the implication does not follow. Take the logic whose sentences consist of a finite sequence of ¬ symbols followed by the symbol ⊥. The theorems are those sentences that have an
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in the article, so I think your formulation (which I reverted) belongs in under that. However, I do not have a good canonical account, so I didn't put it in there because I could do it justice just yet. Be well,
1430:, which however means that our pronouncements can no longer be about completely arbitrary logics. In the following I only use customary fixed interpretations: ⊥ means "false", ⊤ means "true", and ¬ means "not".
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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.
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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).
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Can you give a source for a definition of "tautology" in the sense you understand it, and an example of a tautology that is not a true sentence? Or do you just mean that "A = A" is not true because "A" is a
1982:
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.
2275:
where I’ve highlighted the “if and only if” part, and similarly for the other definitions. As you can see, the equivalence you removed is in a different place than you claim, and is completely harmless.—
489:
If that is a kleene star, I really don't know much about its properties. I cannot vouch for the accuracy of the above statement on those grounds. However, also, Please forgive my error. It's supposed to
382:"Tautology" is the more general term here. All "true sentences" are tautologies, however not all tautologies are "true sentences." That's a good quiz question. It looks like one of the principles of
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How long do you expect me to wait? Given the meaning of completeness that I know and see in the literature, the article is wrong, but you reverted my fix with arguments that I don't comprehend. --
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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.
2333:
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.
156:. 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
2160:, nevertheless your explanation makes no sense at all. The symbolic translation of “A formal system S is semantically complete if and only if every tautology of S is a theorem of S” is
1911:
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
934:
The last of these is particularly interesting because the author uses the terminology "tautological consequence" but makes it explicit in the notation that this corresponds to the
540:
of them. The ink on the paper, or the actual chalk on the board may also be a "tautology." However, it is the properties of the abstract version that we are talking about here.
342:
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
1466:
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. --
2031:
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."
601:
me. I am still waiting for (1) a source for a definition of "tautology" in the sense you understand it, and (2) an example of a tautology that is not a true sentence. --
992:. The syntactic qualities should be covered first, and then the many and varied semantic aspects should supplement. Like I implied earlier, we are both right. Be well,
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iff every sentence that is a tautology is a theorem. (This is from the second definition in that bullet list). There do exist systems that are not
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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.
2264:{\displaystyle \mathrm {Complete} (S){\mathrel {\color {red}\leftrightarrow }}\forall \varphi \,(\models _{S}\varphi \to {}\vdash _{S}\varphi ),}
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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.
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Until or unless a citation for this particular meaning of 'extreme completeness' is forthcoming, this text doesn't belong in the article.
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I see that now the two have been merged. Since I have shown above that the two definitions are not equivalent, the old definition of
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a great deal written about one of the other sorts of completeness in this article that justifies splitting this into two articles.
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2327:, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
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If V is separable, it follows that any vector in V can be written as a (possibly infinite) linear combination of vectors from S.
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the notion for other logical systems than propositional logic. It is hard to imagine a non-semantical method to evaluate
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relate to completeness of a formal system? About to the same extent as it relates to completeness of a dinner set. --
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and not just mechanically scanning for a column of uninterpreted "T" symbols? Even if you say you understand, for all
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Congratulations, you are now a philosopher. I will look into your answerable question at some point! Be well,
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In general they are not equivalent. There is a problem with the notion of "holding universally". You have to
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In the meantime, let me give some references to definitions of soundness and completeness in various texts:
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Then we can do the transition; otherwise, the move may be harmless but is largely a waste of time. IMO. --
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Would it not be a good idea to have
Completeness as a dictionary style definition, leading off to:
2134:{\displaystyle \models _{\mathcal {S}}\varphi \ \leftrightarrow \ \vdash _{\mathcal {S}}\varphi }
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as a sentence, so to speak. To me this seems to be both an unusual and a useless concept. --
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Whether the above can garner consensus or not, there is no impediment to starting a page
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I don't understand your sentence "it is a token or the type of tautology that is ...".
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Last edited at 06:44, 18 April 2008 (UTC). Substituted at 19:53, 1 May 2016 (UTC)
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The formal language with stars and daggers can form theorems, as described at
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Lambiam, the goal here is to account for the properties of the logical system
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The “and only if” should be removed as definitions should use plain “if” per
942:, bolstering my contention that being a tautology is a semantic property. --
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provide the principle that makes me know that it exists: A formal system is
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for the deceptively short disambiguation page for "Completion", mixes
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his term, the standard term should come first, and not vice versa. —
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Handbook of Logic in
Artificial Intelligence and Logic Programming
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yet. The uninterpreted tautology may need elucidation. Be well,
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L, Do you really believe that it is not generally true that:
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It is a theorem that is an easy consequence of axiom A6 of
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Our article now discusses infinite linear combinations. --
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How would you determine whether (for example) the formula
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complete with respect to the property of tautologousness.
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applies only to a rather narrow set of simple logics. --
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complete with respect to the property of tautologousness
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1043:(exdent) May I say that I find the distinction between
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509:"...it is a token of the type OF tautology that is..."
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The main content of this page may need to be moved to
168:) that justifies branching it off to a new article, or
96:, where you can join the project or contribute to the
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are capable of real understanding of the concept of
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is a tautology? The definition given in our article
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220:is a separate article, I'm doing the move now. --
1433:Take the logic whose formal language consists of
86:pages on Knowledge. If you wish to help, you can
76:This disambiguation page is within the scope of
2323:, and are posted here for posterity. Following
2317:The comment(s) below were originally left at
8:
917:Inexhaustibility: A Non-exhaustive Treatment
879:I have no problem with accepting †∗∗†∗ as a
82:, an attempt to structure and organize all
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1899:I removed (twice now) the following text:
1252:can refer to:". Finally, I think the page
1240:, so that the page starts with s.t. like "
1208:is to find stuff here! I strongly propose
1143:pushing symbols around in a neural network
1006:Thank you for your recent contribution to
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1388:and the new one cannot both be right. --
1141:"semantic" interpretation is nothing but
1051:tautology inane? I could likewise define
152:Yes, I agree that this would make a good
92:attached to this talk page, or visit the
32:does not require a rating on Knowledge's
1236:. Furthermore, it should be merged with
386:. That's a quantifier, not some kind of
248:can be written as a (possibly infinite)
193:If and when we move it, it should be to
2219:
905:An Introduction to Substructural Logics
336:without regard to meaning or semantics.
49:
2003:I have my problems with the paragraph
1666:Complete set of commuting observables
1316:This discussion is very old, but the
566:I think the FS system I described at
7:
2027:Definition of an inconsistent system
1010:. I learned something new about it.
528:The things we are talking about are
106:Knowledge:WikiProject Disambiguation
1059:equality. Take an equation such as
109:Template:WikiProject Disambiguation
38:It is of interest to the following
2213:
2191:
2188:
2185:
2182:
2179:
2176:
2173:
2170:
1228:This page should be turned into a
294:Please revert if what I wrote was
14:
2325:several discussions in past years
2206:
1730:Knuth-Bendix completion algorithm
1145:. We are really getting into the
2352:WikiProject Disambiguation pages
1871:yet another kind of completeness
413:I haven't taken a close look at
244:, it follows that any vector in
69:
51:
20:
639:) 15:10, 28 January 2008 (UTC)
310:The subection currently reads:
2255:
2237:
2221:
2207:
2201:
2195:
2111:
1199:This page is crazy! It is the
1:
2293:suggests using a plain "if".
2022:17:05, 23 February 2011 (UTC)
1993:22:32, 12 February 2009 (UTC)
1909:complete in the extreme sense
1851:in mathematics is related to
1577:Complete economic integration
1232:, conforming to the rules of
990:Glossary of first order logic
2303:22:53, 10 January 2013 (UTC)
2283:13:15, 10 January 2013 (UTC)
2053:05:06, 15 January 2012 (UTC)
1977:10:53, 30 January 2009 (UTC)
1961:07:51, 30 January 2009 (UTC)
1944:02:43, 30 January 2009 (UTC)
1929:01:33, 30 January 2009 (UTC)
1720:Gödel's completeness theorem
1230:standard disambiguation page
1185:22:20, 30 January 2008 (UTC)
1163:19:03, 30 January 2008 (UTC)
1105:15:04, 30 January 2008 (UTC)
1089:23:39, 29 January 2008 (UTC)
1020:20:39, 29 January 2008 (UTC)
1002:20:22, 29 January 2008 (UTC)
947:08:43, 29 January 2008 (UTC)
843:00:14, 29 January 2008 (UTC)
690:18:35, 28 January 2008 (UTC)
649:15:10, 28 January 2008 (UTC)
606:12:56, 28 January 2008 (UTC)
580:07:54, 28 January 2008 (UTC)
472:06:32, 28 January 2008 (UTC)
427:00:14, 28 January 2008 (UTC)
374:17:42, 27 January 2008 (UTC)
356:12:30, 27 January 2008 (UTC)
139:Completeness (metric spaces)
2151:23:43, 9 January 2013 (UTC)
2075:23:28, 9 January 2013 (UTC)
1890:04:32, 2 October 2008 (UTC)
1686:Completeness (order theory)
1681:Completeness (cryptography)
1238:Completion (disambiguation)
2368:
2320:Talk:Completeness/Comments
2034:This equivalence holds in
1865:01:13, 29 March 2008 (UTC)
1832:22:59, 28 March 2008 (UTC)
1671:Complete set of invariants
1552:Complete algebraic variety
1471:16:51, 20 March 2008 (UTC)
1393:00:50, 29 March 2008 (UTC)
1378:09:58, 18 March 2008 (UTC)
1362:17:01, 17 March 2008 (UTC)
1304:18:20, 19 March 2008 (UTC)
926:Logic, Sets, and Recursion
391:type of tautology that is:
79:WikiProject Disambiguation
2332:
1858:completeness of a measure
1691:Completeness (statistics)
1456:01:08, 2 April 2008 (UTC)
1415:09:26, 1 April 2008 (UTC)
1346:needs some cleaning up.
1281:08:56, 7 March 2008 (UTC)
1223:08:16, 1 March 2008 (UTC)
326:11:52, 29 Sep 2004 (UTC)
302:06:14, 15 Sep 2003 (UTC)
282:01:57, 15 Sep 2003 (UTC)
273:09:43, 26 Aug 2003 (UTC)
224:23:40 Feb 20, 2003 (UTC)
64:
46:
1700:Completion (ring theory)
1557:Complete bipartite graph
1542:Complete Heyting algebra
1537:Complete Boolean algebra
1462:Please disambiguate this
266:18:12 29 May 2003 (UTC)
213:, Sunday, July 14, 2002
179:, Monday, June 10, 2002
1715:Functional completeness
1517:Command line completion
1442:syntactically complete.
1331:14:35, 4 May 2010 (UTC)
306:Proof theory subsection
142:Completeness (measures)
112:Disambiguation articles
2265:
2135:
2041:principle of explosion
1895:"Extreme Completeness"
1725:Hamiltonian completion
1646:Complete partial order
1497:Bounded complete poset
1273:Complete (mathematics)
893:Michael O'Donnell. In
201:, maybe even complete
2266:
2136:
1740:Model complete theory
1631:Complete metric space
1611:Complete intersection
1547:Complete active space
1532:Complete (complexity)
1321:mathematical sense.
1262:Quadratic reciprocity
830:semantic completeness
447:, where the * is the
2166:
2091:
1853:generalized function
1849:generalized function
1844:generalized function
1775:Strongly NP-complete
1606:Complete information
1294:not helpful at all.
1125:, but how do I know
205:) under the heading
188:Pages_that_link_here
145:Completeness (logic)
1999:Functional Analysis
1903:A formal system is
1837:To the extent that
1780:Turing completeness
1656:Complete quadrangle
1264:currently links to
1256:should be moved to
1147:other minds problem
154:disambiguation page
30:disambiguation page
2313:Assessment comment
2261:
2220:
2210:
2131:
1905:extremely complete
1803:Math v logic again
1785:Weakly NP-complete
1765:Pattern completion
1641:Complete numbering
1527:Complete (box set)
1386:maximally complete
923:Robert L. Causey.
250:linear combination
190:is fixed. -- Anon
34:content assessment
2337:
2336:
2115:
2110:
1936:Pontiff Greg Bard
1824:Pontiff Greg Bard
1799:
1798:
1661:Complete quotient
1572:Complete contract
1567:Complete coloring
1562:Complete category
1428:logical constants
1417:
1405:comment added by
1364:
1352:comment added by
1296:Pontiff Greg Bard
1177:Pontiff Greg Bard
1097:Pontiff Greg Bard
1012:Pontiff Greg Bard
994:Pontiff Greg Bard
835:Pontiff Greg Bard
641:Pontiff Greg Bard
633:Pontiff Greg Bard
572:Pontiff Greg Bard
474:
419:Pontiff Greg Bard
402:(x → x) → (x → x)
363:(p → p) → (p → p)
348:Pontiff Greg Bard
128:
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2127:
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2105:
2104:
2103:
2082:"if and only if"
2077:
1770:Sharp-P-complete
1710:Feature complete
1705:Early completion
1651:Complete protein
1626:Complete measure
1616:Complete lattice
1522:Complete (album)
1476:
1400:
1372:
1347:
1075:tautologies, en
1062:
1008:effective method
530:abstract objects
465:
364:
252:of vectors from
218:Complete measure
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2045:Honestrosewater
2029:
2014:131.234.106.197
2001:
1897:
1873:
1805:
1800:
1790:Word completion
1760:PSPACE-complete
1735:Line completion
1676:Complete theory
1636:Complete mixing
1621:Complete market
1492:Almost complete
1464:
1370:
1343:
1318:well completion
1258:Well completion
1197:
1195:Making it saner
1154:tautology or a
1082:quantifications
1060:
362:
332:
324:Charles Stewart
308:
296:Patent nonsense
207:Generalisations
184:complete_metric
133:
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1596:Complete group
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1591:Complete graph
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1512:Co-RE-complete
1509:
1507:Co-NP-complete
1504:
1502:Chain complete
1499:
1494:
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1399:assumptions?
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1165:
1158:tautology? --
1108:
1107:
1061:5 + 9 = 2 × 7.
1047:tautology and
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1582:Complete game
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1813:mathematics
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1482:AI-complete
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1401:—Preceding
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1077:upside-down
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1266:Completion
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898:pp. 40–41.
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2291:MOS:MATH
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1371:1 < 0
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2238:→
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