|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[mathematical logic]], a [[Theory (mathematical logic)|theory]] is '''complete''' if it is a '''maximal consistent set''' of sentences, i.e., if it is [[consistency|consistent]], and none of its proper extensions is consistent. For theories in logics which contain [[classical logic|classical propositional logic]], this is equivalent to asking that for every [[sentence (mathematical logic)|sentence]] φ in the [[formal language|language]] of the theory it contains either φ itself or its negation ¬φ.
| | The name of the writer is Jayson. To play lacross is some thing he would never give up. I am presently a journey agent. Alaska is exactly where I've usually been living.<br><br>Here is my web page ... [http://clothingcarearchworth.com/index.php?document_srl=441551&mid=customer_review online psychics] |
| | |
| Recursively axiomatizable first-order theories that are rich enough to allow general mathematical reasoning to be formulated cannot be complete, as demonstrated by [[Gödel's incompleteness theorem]].
| |
| | |
| This sense of ''complete'' is distinct from the notion of a complete ''logic'', which asserts that for every theory that can be formulated in the logic, all semantically valid statements are provable theorems (for an appropriate sense of "semantically valid"). [[Gödel's completeness theorem]] is about this latter kind of completeness.
| |
| | |
| Complete theories are closed under a number of conditions internally modelling the [[T-schema]]:
| |
| *For a set <math>S\!</math>: <math>A \land B \in S</math> if and only if <math>A \in S</math> and <math>B \in S</math>,
| |
| *For a set <math>S\!</math>: <math>A \lor B \in S</math> if and only if <math>A \in S</math> or <math>B \in S</math>.
| |
| | |
| Maximal consistent sets are a fundamental tool in the [[model theory]] of [[classical logic]] and [[modal logic]]. Their existence in a given case is usually a straightforward consequence of [[Zorn's lemma]], based on the idea that a [[contradiction]] involves use of only finitely many premises. In the case of modal logics, the collection of maximal consistent sets extending a theory ''T'' (closed under the necessitation rule) can be given the structure of a [[Kripke semantics|model]] of ''T'', called the canonical model.
| |
| | |
| ==Examples==
| |
| Some examples of complete theories are:
| |
| * [[Presburger arithmetic]]
| |
| * [[Tarski's axioms]] for [[Euclidean geometry]]
| |
| * The theory of [[dense linear order]]s
| |
| * The theory of [[algebraically closed field]]s of a given characteristic
| |
| * The theory of [[real closed field]]s
| |
| * Every [[Morley's categoricity theorem|uncountably categorical]] countable theory
| |
| * Every [[omega-categorical theory|countably categorical]] countable theory
| |
| | |
| ==References==
| |
| {{Portal|Logic}}
| |
| * {{cite book |first=Elliott |last=Mendelson |title=Introduction to Mathematical Logic |edition=Fourth edition |year=1997 |publisher=Chapman & Hall |isbn=978-0-412-80830-2| pages=86}}
| |
| | |
| {{Logic}}
| |
| | |
| [[Category:Mathematical logic]]
| |
| [[Category:Model theory]]
| |
| | |
| | |
| {{mathlogic-stub}}
| |
The name of the writer is Jayson. To play lacross is some thing he would never give up. I am presently a journey agent. Alaska is exactly where I've usually been living.
Here is my web page ... online psychics