Forcing isomorphism

1993 ◽  
Vol 58 (4) ◽  
pp. 1291-1301 ◽  
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.

Global quantification in Zermelo-Fraenkel set theory

1985 ◽  
Vol 50 (2) ◽  
pp. 289-301
Author(s):  
John Mayberry
Keyword(s):  
Set Theory ◽  
Order Theory ◽  
Prima Facie ◽  
Local Theory ◽  
Axiom Schema ◽  
First Order ◽  
First Order Theory ◽  
Naturally Occurring ◽  
Universe Of Sets

My aim here is to investigate the role of global quantifiers—quantifiers ranging over the entire universe of sets—in the formalization of Zermelo-Fraenkel set theory. The use of such quantifiers in the formulas substituted into axiom schemata introduces, at least prima facie, a strong element of impredicativity into the thapry. The axiom schema of replacement provides an example of this. For each instance of that schema enlarges the very domain over which its own global quantifiers vary. The fundamental question at issue is this: How does the employment of these global quantifiers, and the choice of logical principles governing their use, affect the strengths of the axiom schemata in which they occur?I shall attack this question by comparing three quite different formalizations of the intuitive principles which constitute the Zermelo-Fraenkel system. The first of these, local Zermelo-Fraenkel set theory (LZF), is formalized without using global quantifiers. The second, global Zermelo-Fraenkel set theory (GZF), is the extension of the local theory obtained by introducing global quantifiers subject to intuitionistic logical laws, and taking the axiom schema of strong collection (Schema XII, §2) as an additional assumption of the theory. The third system is the conventional formalization of Zermelo-Fraenkel as a classical, first order theory. The local theory, LZF, is already very strong, indeed strong enough to formalize any naturally occurring mathematical argument. I have argued (in [3]) that it is the natural formalization of naive set theory. My intention, therefore, is to use it as a standard against which to measure the strength of each of the other two systems.

HODL(ℝ) is a Core Model Below Θ

1995 ◽  
Vol 1 (1) ◽  
pp. 75-84 ◽  
Author(s):  
John R. Steel
Keyword(s):  
Set Theory ◽  
Transitive Closure ◽  
Order Theory ◽  
First Order ◽  
First Order Theory ◽  
Inner Model ◽  
Definable Sets ◽  
The Axiom Of Choice ◽  
The Universe ◽  
Do So

In this paper we shall answer some questions in the set theory of L(ℝ), the universe of all sets constructible from the reals. In order to do so, we shall assume ADL(ℝ), the hypothesis that all 2-person games of perfect information on ω whose payoff set is in L(ℝ) are determined. This is by now standard practice. ZFC itself decides few questions in the set theory of L(ℝ), and for reasons we cannot discuss here, ZFC + ADL(ℝ) yields the most interesting “completion” of the ZFC-theory of L(ℝ).ADL(ℝ) implies that L(ℝ) satisfies “every wellordered set of reals is countable”, so that the axiom of choice fails in L(ℝ). Nevertheless, there is a natural inner model of L(ℝ), namely HODL(ℝ), which satisfies ZFC. (HOD is the class of all hereditarily ordinal definable sets, that is, the class of all sets x such that every member of the transitive closure of x is definable over the universe from ordinal parameters (i.e., “OD”). The superscript “L(ℝ)” indicates, here and below, that the notion in question is to be interpreted in L(R).) HODL(ℝ) is reasonably close to the full L(ℝ), in ways we shall make precise in § 1. The most important of the questions we shall answer concern HODL(ℝ): what is its first order theory, and in particular, does it satisfy GCH?These questions first drew attention in the 70's and early 80's. (See [4, p. 223]; also [12, p. 573] for variants involving finer notions of definability.)

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

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.

scholarly journals Nonexistence of universal orders in many cardinals

1992 ◽  
Vol 57 (3) ◽  
pp. 875-891 ◽  
Author(s):  
Menachem Kojman ◽  
Saharon Shelah
Keyword(s):  
Linear Order ◽  
Order Theory ◽  
Order Property ◽  
Universal Model ◽  
Covering Theorem ◽  
First Order ◽  
First Order Theory ◽  
Ordered Fields ◽  
Singular Cardinals ◽  
Universal Order

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).

Complete theories with only universal and existential axioms

1987 ◽  
Vol 52 (3) ◽  
pp. 698-711 ◽  
Author(s):  
A. H. Lachlan
Keyword(s):  
Order Theory ◽  
First Order ◽  
First Order Theory ◽  
Existential Sentences ◽  
Complete Theories ◽  
The Universe

AbstractLet T be a complete first-order theory over a finite relational language which is axiomatized by universal and existential sentences. It is shown that T is almost trivial in the sense that the universe of any model of T can be written . where F is finite and I1, I2, …, In are mutually indiscernible over F. Some results about complete theories with ∃∀-axioms over a finite relational language are also mentioned.

scholarly journals On equations and first-order theory of one-relator monoids

2021 ◽  
pp. 104745
Author(s):  
Albert Garreta ◽  
Robert D. Gray
Keyword(s):  
Order Theory ◽  
First Order ◽  
First Order Theory

scholarly journals A first-order theory of Ulm type

Computability ◽  
2019 ◽  
Vol 8 (3-4) ◽  
pp. 347-358
Author(s):  
Matthew Harrison-Trainor
Keyword(s):  
Order Theory ◽  
First Order ◽  
First Order Theory

scholarly journals Quasi-decidability of a Fragment of the First-Order Theory of Real Numbers

2015 ◽  
Vol 57 (2) ◽  
pp. 157-185 ◽  
Author(s):  
Peter Franek ◽  
Stefan Ratschan ◽  
Piotr Zgliczynski
Keyword(s):  
Order Theory ◽  
Real Numbers ◽  
First Order ◽  
First Order Theory
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