Two structures M and N of the same signature σ are elementarily equivalent if every first-order sentence (formula without free variables) over σ is true in M if and only if it is true in N, i.e. if M and N have the same complete first-order theory. If M and N are elementarily equivalent, one writes M ≡ N. A first-order theory is complete if and only if any two of its models are elementarily equivalent. For example, consider the language with one binary relation symbol '<'.
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The back-and-forth method can be used to show that any two countable atomic models of a theory that are elementarily equivalent are isomorphic.
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The model R of real numbers with its usual order and the model Q of rational numbers with its usual order are elementarily equivalent, since they both interpret '<' as an unbounded dense linear ordering. This is sufficient to ensure elementary equivalence, because the theory of unbounded dense linear orderings is complete, as can be shown by the Łoś–Vaught test. More generally, any first-order theory with an infinite model has non- isomorphic, elementarily equivalent models, which can be obtained via the Löwenheim–Skolem theorem. Thus, for example, there are non-standard models of Peano arithmetic, which contain other objects than just the numbers 0, 1, 2, etc.
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In mathematics, extendible cardinals are large cardinals introduced by , who was partly motivated by reflection principles. Intuitively, such a cardinal represents a point beyond which initial pieces of the universe of sets start to look similar, in the sense that each is elementarily embeddable into a later one.
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Fraïssé used the back-and- forth method to determine whether two model-theoretic structures were elementarily equivalent. This method of determining elementary equivalence was later formulated as the Ehrenfeucht–Fraïssé game. Fraïssé worked primarily in relation theory. Another of his important works was the Fraïssé construction of a Fraïssé limit of finite structures.
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In mathematics, and in particular model theory, a prime model is a model that 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).
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An example, important in field theory, are irreducible polynomials, i.e., irreducible elements in k[X], for a field k. The fact that Z is a UFD can be stated more elementarily by saying that any natural number can be uniquely decomposed as product of powers of prime numbers. It is also known as the fundamental theorem of arithmetic.
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It is known that the complete theory T of an o-minimal structure is an o-minimal theory.Knight, Pillay and Steinhorn (1986), Pillay and Steinhorn (1988). This result is remarkable because, in contrast, the complete theory of a minimal structure need not be a strongly minimal theory, that is, there may be an elementarily equivalent structure which is not minimal.
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Sela's work relied on applying his earlier JSJ-decomposition and real tree techniques as well as developing new ideas and machinery of "algebraic geometry" over free groups. Sela pushed this work further to study first-order theory of arbitrary torsion-free word-hyperbolic groups and to characterize all groups that are elementarily equivalent to (that is, have the same first-order theory as) a given torsion-free word-hyperbolic group.
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In model theory, a branch of mathematical logic, two structures M and N of the same signature σ are called elementarily equivalent if they satisfy the same first-order σ-sentences. If N is a substructure of M, one often needs a stronger condition. In this case N is called an elementary substructure of M if every first-order σ-formula φ(a1, …, an) with parameters a1, …, an from N is true in N if and only if it is true in M. If N is an elementary substructure of M, then M is called an elementary extension of N. An embedding h: N → M is called an elementary embedding of N into M if h(N) is an elementary substructure of M. A substructure N of M is elementary if and only if it passes the Tarski–Vaught test: every first-order formula φ(x, b1, …, bn) with parameters in N that has a solution in M also has a solution in N when evaluated in M. One can prove that two structures are elementarily equivalent with the Ehrenfeucht–Fraïssé games.
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In particular, his work implies that if a finitely generated group G is elementarily equivalent to a word- hyperbolic group then G is word-hyperbolic as well. Sela also proved that the first-order theory of a finitely generated free group is stable in the model- theoretic sense, providing a brand-new and qualitatively different source of examples for the stability theory. An alternative solution for the Tarski conjecture has been presented by Olga Kharlampovich and Alexei Myasnikov.O. Kharlampovich, and A. Myasnikov.
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A field containing is called an algebraic closure of if it is algebraic over (roughly speaking, not too big compared to ) and is algebraically closed (big enough to contain solutions of all polynomial equations). By the above, is an algebraic closure of . The situation that the algebraic closure is a finite extension of the field is quite special: by the Artin-Schreier theorem, the degree of this extension is necessarily 2, and is elementarily equivalent to . Such fields are also known as real closed fields.
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In mathematical logic, an elementary definition is a definition that can be made using only finitary first-order logic, and in particular without reference to set theory or using extensions such as plural quantification. Elementary definitions are of particular interest because they admit a complete proof apparatus while still being expressive enough to support most everyday mathematics (via the addition of elementarily-expressible axioms such as Zermelo–Fraenkel set theory (ZFC)). Saying that a definition is elementary is a weaker condition than saying it is algebraic.
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In contrast with the notion of saturated model, prime models are restricted to very specific cardinalities by the Löwenheim–Skolem theorem. If L is a first-order language with cardinality \kappa and T is a complete theory over L, then this theorem guarantees a model for T of cardinality \max(\kappa,\aleph_0). Therefore no prime model of T can have larger cardinality since at the very least it must be elementarily embedded in such a model. This still leaves much ambiguity in the actual cardinality.
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The main idea behind the game is that we have two structures, and two players (defined below). One of the players wants to show that the two structures are different, whereas the other player wants to show that they are elementarily equivalent (satisfy the same first-order sentences). The game is played in turns and rounds. A round proceeds as follows: the first player (the spoiler) first chooses any element from one (either) of the structures, and the second player (the duplicator) chooses an element from the other structure.
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This second approach is sometimes called first order logic with equality, but many authors adopt it for the general study of first-order logic without comment. There are a few other reasons to restrict study of first-order logic to normal models. First, it is known that any first-order interpretation in which equality is interpreted by an equivalence relation and satisfies the substitution axioms for equality can be cut down to an elementarily equivalent interpretation on a subset of the original domain. Thus there is little additional generality in studying non-normal models.
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In model theory, there are several general results and definitions related to absoluteness. A fundamental example of downward absoluteness is that universal sentences (those with only universal quantifiers) that are true in a structure are also true in every substructure of the original structure. Conversely, existential sentences are upward absolute from a structure to any structure containing it. Two structures are defined to be elementarily equivalent if they agree about the truth value of all sentences in their shared language, that is, if all sentences in their language are absolute between the two structures.
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In the mathematical discipline of model theory, the Ehrenfeucht–Fraïssé game (also called back-and-forth games) is a technique for determining whether two structures are elementarily equivalent. The main application of Ehrenfeucht–Fraïssé games is in proving the inexpressibility of certain properties in first-order logic. Indeed, Ehrenfeucht–Fraïssé games provide a complete methodology for proving inexpressibility results for first-order logic. In this role, these games are of particular importance in finite model theory and its applications in computer science (specifically computer aided verification and database theory), since Ehrenfeucht–Fraïssé games are one of the few techniques from model theory that remain valid in the context of finite models.
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Kc computes the successors of measurable and many singular cardinals correctly. Also, it is expected that under an appropriate weakening of countable certifiability, Kc would correctly compute the successors of all weakly compact and singular strong limit cardinals correctly. If V is closed under a mouse operator (an inner model operator), then so is Kc. Kc has no sharp: There is no natural non-trivial elementary embedding of Kc into itself. (However, unlike K, Kc may be elementarily self-embeddable.) If in addition there are also no Woodin cardinals in this model (except in certain specific cases, it is not known how the core model should be defined if Kc has Woodin cardinals), we can extract the actual core model K. K is also its own core model.
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The notion of saturated model is dual to the notion of prime model in the following way: let T be a countable theory in a first-order language (that is, a set of mutually consistent sentences in that language) and let P be a prime model of T. Then P admits an elementary embedding into any other model of T. The equivalent notion for saturated models is that any "reasonably small" model of T is elementarily embedded in a saturated model, where "reasonably small" means cardinality no larger than that of the model in which it is to be embedded. Any saturated model is also homogeneous. However, while for countable theories there is a unique prime model, saturated models are necessarily specific to a particular cardinality. Given certain set-theoretic assumptions, saturated models (albeit of very large cardinality) exist for arbitrary theories.
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The first incompleteness theorem states that for any consistent, effectively given (defined below) logical system that is capable of interpreting arithmetic, there exists a statement that is true (in the sense that it holds for the natural numbers) but not provable within that logical system (and which indeed may fail in some non-standard models of arithmetic which may be consistent with the logical system). For example, in every logical system capable of expressing the Peano axioms, the Gödel sentence holds for the natural numbers but cannot be proved. Here a logical system is said to be effectively given if it is possible to decide, given any formula in the language of the system, whether the formula is an axiom, and one which can express the Peano axioms is called "sufficiently strong." When applied to first-order logic, the first incompleteness theorem implies that any sufficiently strong, consistent, effective first-order theory has models that are not elementarily equivalent, a stronger limitation than the one established by the Löwenheim–Skolem theorem.
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N is an elementary substructure of M if N and M are structures of the same signature σ such that for all first- order σ-formulas φ(x1, …, xn) with free variables x1, …, xn, and all elements a1, …, an of N, φ(a1, …, an) holds in N if and only if it holds in M: :N \models φ(a1, …, an) iff M \models φ(a1, …, an). It follows that N is a substructure of M. If N is a substructure of M, then both N and M can be interpreted as structures in the signature σN consisting of σ together with a new constant symbol for every element of N. Then N is an elementary substructure of M if and only if N is a substructure of M and N and M are elementarily equivalent as σN-structures. If N is an elementary substructure of M, one writes N \preceq M and says that M is an elementary extension of N: M \succeq N. The downward Löwenheim–Skolem theorem gives a countable elementary substructure for any infinite first-order structure in at most countable signature; the upward Löwenheim–Skolem theorem gives elementary extensions of any infinite first-order structure of arbitrarily large cardinality.
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