| ||
Model theory(modeltheory)In mathematics , model theory is the study of therepresentation of mathematical concepts in terms of set theory , or the study ofthe models which underlie mathematical systems . It assumesthat there are some pre-existing mathematical objects out there, and asks questions regarding how or what can be proven given theobjects, some operations or relations amongst the objects, and a set of axioms. The independence of the axiom of choice and the continuum hypothesis from the other axioms of set theory (proven by Paul Cohen and Kurt Gödel ) are the two most famous results arising from model theory. It wasproven that both the axiom of choice and its negation are consistent with the Zermelo-Fraenkel axioms of set theory; the same result holds for the continuum hypothesis. Theseresults are a part of axiomatic set theory , a particularapplication of model theory. An example of the concepts of model theory is provided by the theory of the real numbers . We start with a set of individuals, where each individual is a real number, and a set ofrelations and/or functions, such as { ×, +, −, . , 0, 1 }. If we ask a question such as "∃ y (y ×y = 1 + 1)" in this language, then it is clear that the sentence is true for the reals - there is such a real numbery, namely the square root of 2; for the rational numbers , however, the sentence is false. Conversely, "∃y (y × y = 0 − 1)" is false in the reals - to make it true we can add a constant symboli and a new axiom "i × i = 0 − 1", which gives us the complex numbers . Model theory is then concerned with what is provable within given mathematical systems, and how these systems relate to eachother. It is particularly concerned with what happens when we try to extend some system by the addition of new axioms or newlanguage constructs. A model is formally defined in context of some language L. . The model consists of two things:
A theory is defined as a set of sentences which is consistent; often it is also defined to be closed under logicalconsequence . For example, the set of all sentences true in some particular model (e.g. the reals) is a theory. Gödel's completeness theorem saysthat a theory has a model if and only if it is consistent , i.e. nocontradiction is proved by the theory. This is the heart of model theory as it lets us answer questions about theories by lookingat models and vice-versa. One should not confuse the completeness theorem with the notion of a complete theory . A complete theoryis a theory which contains every sentence or it's negation. Importantly one can find a complete consistent theoryextending any consistent theory. The compactness theorem states that a set of sentences Sis satisfiable, i.e., has a model, if every finite subset of S is satisfiable. In the context of proof theory the analogousstatement is trivial, since every proof can have only a finite number of antecedents used in the proof; in the context of modeltheory however, this proof is somewhat more difficult. There are two well known proofs, one by Gödel (which goes via proofs) and one by Malcev (which is more direct and allows us to restrict the cardinality of the resulting model). Model theory is usually concerned with first order logic andmany important results (such as the completeness and compactness theorems) fail in secondorder logic or other alternatives. In first order logic all infinite cardinals look the same to a language which is countable . This is expressed in the Löwenheim-Skolem theorems - which state that any theory with an infinite model A hasmodels of all infinite cardinalities (at least that of the language) which agree with A on all sentences - they are"elementarily equivalent". So in particular, set theory (whose language is countable ) has a countable model - this is known as Skolem's Paradox, even though it's true (providing youaccept the axioms of set theory)! To see why it was thought paradoxical , considerthat there are sentences in set theory which postulate the existence of uncountable sets - and these sentences are true in ourcountable model. Particularly the proof of the independence of the hypothesis requires considering sets in models which appear tobe uncountable when viewed from within the model, but are countable to someone outside the model. TODO - Vaught's test. Extensions, Embeddings and Diagrams. To give a flavor, mentioning the hyperreals and/or the extension of the concepts of basis and dimension to strongly minimal theories would begood. (All of these need substantial filling out) Note: The unrelated term ' mathematical model ' isalso used informally in other parts of mathematics and science. See also
, proof, model thoery, sentences, model theoy, mathematical, modle theory, consistent, model theori, one, model teory, two, model hteory, mathematics, model tehory, axiom, moel theory, defined, model theor, infinite, mdel theory, numbers, modeltheory, models, model theroy, sentence, mdoel theory, every, model theoyr, logic, omdel theory, real, mode theory, complete, mode ltheory, false, model thory, constant, odel theory, particular, moedl theory, order, model thery, finite, model heory, proofs, modl theory, theorems, modelt heory, domain This article is completely or partly from Wikipedia - The Free Online Encyclopedia. Original Article. The text on this site is made available under the terms of the GNU Free Documentation Licence. We take no responsibility for the content, accuracy and use of this article. Anoca.org Encyclopedia 0.02s |