Nonexistence of universal orders in many cardinals

AbstractOur theme is that not every interesting question in set theory is independent of ZFC. We give an example of a first order theory T with countable D(T) which cannot have a universal model at ℵ1; without CH; we prove in ZFC a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove—again in ZFC—that for a large class of cardinals there is no universal linear order (e.g. in every regular ). In fact, what we show is that if there is a universal linear order at a regular λ and its existence is not a result of a trivial cardinal arithmetical reason, then λ “resembles” ℵ1—a cardinal for which the consistency of having a universal order is known. As for singular cardinals, we show that for many singular cardinals, if they are not strong limits then they have no universal linear order. As a result of the nonexistence of a universal linear order, we show the nonexistence of universal models for all theories possessing the strict order property (for example, ordered fields and groups, Boolean algebras, p-adic rings and fields, partial orders, models of PA and so on).

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More on definable sets of p-adic numbers

Journal of Symbolic Logic ◽

10.2307/2274581 ◽

1988 ◽

Vol 53 (3) ◽

pp. 912-920 ◽

Cited By ~ 2

Author(s):

Philip Scowcroft

Keyword(s):

Cross Section ◽

Quantifier Elimination ◽

Order Theory ◽

Model Completeness ◽

First Order ◽

First Order Theory ◽

Ordered Fields ◽

Definable Sets ◽

The Cross ◽

Primitive Recursive

To eliminate quantifiers in the first-order theory of the p-adic field Qp, Ax and Kochen use a language containing a symbol for a cross-section map n → pn from the value group Z into Qp [1, pp. 48–49]. The primitive-recursive quantifier eliminations given by Cohen [2] and Weispfenning [10] also apply to a language mentioning the cross-section, but none of these authors seems entirely happy with his results. As Cohen says, “all the operations… introduced for our simple functions seem natural, with the possible exception of the map n → pn” [2, p. 146]. So all three authors show that various consequences of quantifier elimination—completeness, decidability, model-completeness—also hold for a theory of Qp not employing the cross-section [1, p. 453; 2, p. 146; 10, §4]. Macintyre directs a more specific complaint against the cross-section [5, p. 605]. Elementary formulae which use it can define infinite discrete subsets of Qp; yet infinite discrete subsets of R are not definable in the language of ordered fields, and so certain analogies between Qp and R suggested by previous model-theoretic work seem to break down.To avoid this problem, Macintyre gives up the cross-section and eliminates quantifiers in a theory of Qp written just in the usual language of fields supplemented by a predicate V for Qp's valuation ring and by predicates Pn for the sets of nth powers in Qp (for all n ≥ 2).

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Probabilities of first-order sentences on sparse random relational structures: an application to definability on random CNF formulas

Journal of Logic and Computation ◽

10.1093/logcom/exaa061 ◽

2020 ◽

Author(s):

Lázaro Alberto Larrauri

Keyword(s):

Random Graphs ◽

Order Theory ◽

Boolean Satisfiability ◽

Random Model ◽

Order Property ◽

Expected Number ◽

First Order ◽

Relational Structures ◽

First Order Theory ◽

Cnf Formulas

Abstract We extend the convergence law for sparse random graphs proven by Lynch to arbitrary relational languages. We consider a finite relational vocabulary $\sigma $ and a first-order theory $T$ for $\sigma $ composed of symmetry and anti-reflexivity axioms. We define a binomial random model of finite $\sigma $-structures that satisfy $T$ and show that first-order properties have well defined asymptotic probabilities when the expected number of tuples satisfying each relation in $\sigma $ is linear. It is also shown that these limit probabilities are well behaved with respect to several parameters that represent the density of tuples in each relation $R$ in the vocabulary $\sigma $. An application of these results to the problem of random Boolean satisfiability is presented. We show that in a random $k$-CNF formula on $n$ variables, where each possible clause occurs with probability $\sim c/n^{k-1}$, independently any first-order property of $k$-CNF formulas that implies unsatisfiability does almost surely not hold as $n$ tends to infinity.

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Dependent theories and the generic pair conjecture

Communications in Contemporary Mathematics ◽

10.1142/s0219199715500042 ◽

2014 ◽

Vol 17 (01) ◽

pp. 1550004 ◽

Cited By ~ 2

Author(s):

Saharon Shelah

Keyword(s):

Linear Order ◽

Order Theory ◽

Saturated Model ◽

First Order ◽

First Order Theory ◽

Dependent Theory ◽

The One ◽

Decomposition Theorems ◽

Measurable Cardinals

We try to understand complete types over a somewhat saturated model of a complete first-order theory which is dependent (previously called NIP), by "decomposition theorems for such types". Our thesis is that the picture of dependent theory is the combination of the one for stable theories and the one for the theory of dense linear order or trees (and first, we should try to understand the quite saturated case). As a measure of our progress, we give several applications considering some test questions; in particular, we try to prove the generic pair conjecture and do it for measurable cardinals.

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Forcing isomorphism

Journal of Symbolic Logic ◽

10.2307/2275144 ◽

1993 ◽

Vol 58 (4) ◽

pp. 1291-1301 ◽

Cited By ~ 1

Author(s):

J. T. Baldwin ◽

M. C. Laskowski ◽

S. Shelah

Keyword(s):

Partially Ordered Set ◽

Ordered Set ◽

Order Theory ◽

Dependence Structure ◽

Order Property ◽

First Order ◽

First Order Theory ◽

Omitting Types ◽

Universe Of Sets ◽

The Universe

If two models of a first-order theory are isomorphic, then they remain isomorphic in any forcing extension of the universe of sets. In general however, such a forcing extension may create new isomorphisms. For example, any forcing that collapses cardinals may easily make formerly nonisomorphic models isomorphic. However, if we place restrictions on the partially-ordered set to ensure that the forcing extension preserves certain invariants, then the ability to force nonisomorphic models of some theory T to be isomorphic implies that the invariants are not sufficient to characterize the models of T.A countable first-order theory is said to be classifiable if it is superstable and does not have either the dimensional order property (DOP) or the omitting types order property (OTOP). If T is not classifiable, Shelah has shown in [5] that sentences in L∞,λ do not characterize models of T of power λ. By contrast, in [8] Shelah showed that if a theory T is classifiable, then each model of cardinality λ is described by a sentence of L∞,λ. In fact, this sentence can be chosen in the . ( is the result of enriching the language by adding for each μ < λ a quantifier saying the dimension of a dependence structure is greater than μ) Further work ([3], [2]) shows that ⊐+ can be replaced by ℵ1.

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A generalized model companion for a theory of partially ordered fields

Journal of Symbolic Logic ◽

10.2307/2273301 ◽

1979 ◽

Vol 44 (4) ◽

pp. 643-652

Author(s):

Werner Stegbauer

Keyword(s):

General Model ◽

Order Theory ◽

Model Companion ◽

Generalized Model ◽

First Order ◽

First Order Theory ◽

Ordered Fields ◽

Closed Fields ◽

Algebraically Closed Fields ◽

Model Completion

The notion of a model companion for a first-order theory T was introduced and discussed in [1] and [2] as a generalization of the concept of a model completion of a theory. Both concepts reflect, on a general model theoretic level, properties of the theory of algebraically closed fields. The literature provides many examples of first-order theories with and without model companions—see [3] for a survey of these results. In this paper, we give a further generalization of the notion of a model companion.Roughly speaking, we allow instead of embeddings more general classes of maps (e.g. homomorphisms) and we allow any set of formulas which is preserved by these maps instead of existential formulas. This plan is worked out in detail in [5], where we discuss also several examples. One of these examples is given in this paper.In order to clarify the model theoretic background, we now introduce the relevant concepts and theorems from [5].

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