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First-order Model Theory (Stanford Encyclopedia of Philosophy) Elementary equivalence is an equivalence relation on the class of all L-structures. This description implied among other things that the structures elementarily equivalent to R are exactly the real-closed fields, a class of fields which was already known to the algebraists in its own right. Isomorphism is an equivalence relation on the class of all structures of a fixed signature K. If two structures are isomorphic then they share all model-theoretic properties; in particular they are elementarily equivalent. www.seop.leeds.ac.uk /entries/modeltheory-fo   (6181 words)

Elementarily equivalent - Wikipedia, the free encyclopedia In mathematics, specifically model theory, two models of a language are said to be elementarily equivalent if their theories are the same; that is, any sentence satisfied by one model is also satisfied by the other. The model R of real numbers and the model Q of rational numbers are elementarily equivalent, since they both translate '<' as an unbounded dense linear ordering. There also exist non-standard models of number theory, which contain other objects than just the numbers 0, 1, 2, etc. However, the language is the same as standard number theory, since these extra objects cannot be mentioned. en.wikipedia.org /wiki/Elementarily_equivalent   (150 words)

Prime model - Wikipedia, the free encyclopedia In mathematics, and in particular model theory, a prime model is a model which is as simple as possible. Specifically, a model P is prime if it admits an elementary embedding into any model M to which it is elementarily equivalent (that is, into any model M satisfying the same complete theory as P). In contrast with the notion of saturated model, prime models are restricted to very specific cardinalities by the Löwenheim-Skolem theorem. en.wikipedia.org /wiki/Prime_model   (431 words)

First-order Model Theory Elementary equivalence is an equivalence relation on the class of all L-structures. This description implied among other things that the structures elementarily equivalent to R are exactly the real-closed fields, a class of fields which was already known to the algebraists in its own right. Isomorphism is an equivalence relation on the class of all structures of a fixed signature K. If two structures are isomorphic then they share all model-theoretic properties; in particular they are elementarily equivalent. plato.stanford.edu /entries/modeltheory-fo   (6269 words)

INI : Abstracts : MAAW01 : Some results in elementary equivalence of linear and algebraic groups and other structures He proved that the group G(m,K)$ is elementarily equivalent to the group $G(n,L) (G=GL,PGL,SL,PSL, m,n>2, K and L are fields of characteristic~0} iff m=n and the fields K and L are elementarily equivalent. Then g and G' are elementarily equivalent iff K and K' are elementarily equivalent and the algebras L and L' are isomorphic. For example, they proved that two categories mod-R and mod-R' (R and R' are commutative, local, semilocal, Artinean rings) are elementarily equivalent iff the rings R and R' are equivalent in the second order logic. www.newton.cam.ac.uk /programmes/MAA/Poster1/bunina.html   (431 words)

Antimeta My good friend Dan Korman from UT Austin has now joined the blog at Close Range, starting with a provocative post arguing that the traditional analysis of "size of a set" as "equivalence class under bijections (one-one mappings)" is incorrect. I'd like to suggest that there is no fact of the matter as to what the "size of a set" is, and that our intuitions correspond to multiple distinct notions, rather than just one notion as Dan (and those who argue in favor of the bijection analysis) suggest. He can't mean just that every sentence true of the elements of one set is true of the elements of the other (i.e., that they are elementarily equivalent as models), because the Löwenheim-Skolem Theorem guarantees that for every infinite set, we can find an elementarily equivalent model of every infinite cardinality. www.ocf.berkeley.edu /~easwaran/blog   (2530 words)

Model theory Model theory is concerned with first order logic, and to first order logic all infinite cardinals look the same. This is expressed in the Löwenheim-Skolem theorems - which state that any theory with an infinite model A has models 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 you accept the axioms of set theory)! www.sciencedaily.com /encyclopedia/model_theory   (826 words)

On LaRouche’s Discovery Once that student has adduced a sense of the equivalence (higher hypothesis) of valid past discoveries of an axiomatic-revolutionary quality, the student's first resort, at each confronting of an unfamiliar such discovery, is to test that discovery for its quality of Cantorian equivalence. The equivalence among past discoveries (hypotheses) reflects the test of an implicit increase of mankind's potential population-density. That increasing is the equivalence of the higher hypothesis as itself a process. www.schillerinstitute.org /fid_91-96/941_lar_discovery-2.html   (8603 words)

entertainment.ca - elementarily equivalent   (Site not responding. Last check: 2007-10-12) In some cases, even more is true: a theory is complete if all its models are elementarily equivalent (the same sentences are true in each). We define an equivalence relation @ º and, @ is elementarily equivalent (EE) to and, if for every sentence f of L, @ = f ß à and = f. Shelah's theorem, saying that two structures are elementarily equivalent if and only if they have isomorphic ultrapowers. entertainment.ca /elementarily-equivalent/reference/search   (72 words)

[No title] Then ${\cal M}^+$ is said to be {\em ${\cal M}$-minimal} if, for every ${\cal N}^+$ elementarily equivalent to ${\cal M}^+$, every parameter-definable subset of its domain $N^+$ is definable with parameters by a quantifier-free $L$-formula. We say that ${\cal F}$ is {\em $P$-minimal} if, for every ${\cal F'}$ elementarily equivalent to ${\cal F}$, every definable subset of $F'$ is quantifier-free definable by an $L_d$-formula. Condition (ii) is equivalent to the condition that every non-empty definable subset of the value group which is bounded below has a least element. www.amsta.leeds.ac.uk /Pure/preprints/hdm/hdm8   (4230 words)

[No title] Turing and enumeration degrees Marat M. Arslanov Kazan State University The well-known problem of the elementary equivalence of the partial orderings of the n-c.e. by the same formula) definable in the partial orderings D_{2n}, and D_{2n+1}, then they are not elementarily equivalent. Turing degrees elementarily equivalent for n \not= m if there is a unary predicate for the c.e. www.math.psu.edu /simpson/talks/cta/arsl   (262 words)

Citations: Every two elementarily equivalent models have isomorphic ultrapowers - Shelah (ResearchIndex) One cannot replace 9 1 equivalence by even 89 equivalence in general as the.... Shelah, Every two elementarily equivalent models have isomorphic ultrapowers, Israel J. Math 10 (1971), 224 233. So to prove that a class is non elementary, it suces to show that it is not closed under either or both.... citeseer.ist.psu.edu /context/1308213/0   (525 words)

Learn more about Model theory in the online encyclopedia.   (Site not responding. Last check: 2007-10-12) Model theory is concerned with first order logic, and to first order logic all cardinals look the same. This is expressed in the Lowenheim-Skolem theorems - which state that any theory with an infinite model A has models 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! www.onlineencyclopedia.org /m/mo/model_theory.html   (805 words)

CrunchyLogic » Mathematical Logic Observe that if M and M’ are isomorphic, then they are elementarily equivalent, though the converse does not hold (the canonical, no pun intended, example being the non-standard models of arithmetic.) are both κ-sized models of T, yet are not elementarily equivalent (due to disagreement on truth of θ) and therefore not isomorphic, contradicting our assumption that T is κ-categorical. This theorem is sometimes given with as many as 7 equivalent conditions–too much for me. See a real mathematician for other formulations. www.underlevel.net /crunchylogic/?cat=10   (933 words)

The dreams that stuff is made of » 2007 » January Let me try to define in very precise terms what elementary compositionality is. First, let’s define the language of a rule to be the set of all strings that it matches exactly. Then, to borrow a term from model theory, two rules are elementarily equivalent if they have exactly the same language. Stated less formally: if you replace every call to a rule with a call to an elementarily equivalent rule, nothing changes. luqui.org /blog/archives/2007/01   (5044 words)

Citations: Equations and inequations on finite and infinite trees - Colmerauer (ResearchIndex) FT is complete, all three models are elementarily equivalent (i.e. Our proof of FT s completeness will exhibit a simplification algorithm that computes for every feature description an equivalent solved form from which the solutions of the description can be read of easily. Thus, under the assumption of independence, deciding entailment is equivalent to solving conjunctions of positive and negative constraints. citeseer.ist.psu.edu /context/62339/0   (2307 words)

Citebase - Almost locally free groups and the genus question   (Site not responding. Last check: 2007-10-12) It is shown that in every model of Th(F$_2$)$\cap \forall \exists $ the maximal Abelian subgroups are elementarily equivalent to locally cyclic groups (necessarily nontrivial and torsion free). Two classes of groups are interpolated between the non-Abelian locally free groups and Remeslennikov's $\exists $-free groups. In particular, the almost locally free% \textbf{\ }groups are the models of Th(F$_2$)$\cap \forall \exists $ while the quasi-locally free groups are the $\exists $-free groups with maximal Abelian subgroups elemenatarily equivalent to locally cyclic groups (necessarily nontrivial and torsion free). citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/9603204   (588 words)

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