Models with second order properties in successors of singulars

1989 ◽  
Vol 54 (1) ◽  
pp. 122-137
Author(s):  
Rami Grossberg
Keyword(s):  
Order Theory ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Complete Theory ◽  
Singular Cardinal ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
First Order ◽  
First Order Theory

AbstractLet L(Q) be first order logic with Keisler's quantifier, in the λ+ interpretation (= the satisfaction is defined as follows: M ⊨ (Qx)φ(x) means there are λ+ many elements in M satisfying the formula φ(x)).Theorem 1. Let λ be a singular cardinal; assume □λ and GCH. If T is a complete theory in L(Q) of cardinality at most λ, and p is an L(Q) 1-type so that T strongly omits p( = p has no support, to be defined in §1), then T has a model of cardinality λ+ in the λ+ interpretation which omits p.Theorem 2. Let λ be a singular cardinal, and let T be a complete first order theory of cardinality λ at most. Assume □λ and GCH. If Γ is a smallness notion then T has a model of cardinality λ+ such that a formula φ(x) is realized by λ+ elements of M iff φ(x) is not Γ-small. The theorem is proved also when λ is regular assuming λ = λ<λ. It is new when λ is singular or when ∣T∣ = λ is regular.Theorem 3. Let λ be singular. If Con(ZFC + GCH + ∃κ) [κ is a strongly compact cardinal]), then the following is consistent: ZFC + GCH + the conclusions of all above theorems are false.

scholarly journals GEOMETRISATION OF FIRST-ORDER LOGIC

2015 ◽  
Vol 21 (2) ◽  
pp. 123-163 ◽  
Author(s):  
ROY DYCKHOFF ◽  
SARA NEGRI
Keyword(s):  
Proof Theory ◽  
Normal Forms ◽  
Order Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Conservative Extension ◽  
First Order ◽  
First Order Theory ◽  
Intermediate Logics ◽  
Normal Form Theorem

AbstractThat every first-order theory has a coherent conservative extension is regarded by some as obvious, even trivial, and by others as not at all obvious, but instead remarkable and valuable; the result is in any case neither sufficiently well-known nor easily found in the literature. Various approaches to the result are presented and discussed in detail, including one inspired by a problem in the proof theory of intermediate logics that led us to the proof of the present paper. It can be seen as a modification of Skolem’s argument from 1920 for his “Normal Form” theorem. “Geometric” being the infinitary version of “coherent”, it is further shown that every infinitary first-order theory, suitably restricted, has a geometric conservative extension, hence the title. The results are applied to simplify methods used in reasoning in and about modal and intermediate logics. We include also a new algorithm to generate special coherent implications from an axiom, designed to preserve the structure of formulae with relatively little use of normal forms.

Topos-theoretic background

2018 ◽  
Author(s):  
Olivia Caramello
Keyword(s):  
Category Theory ◽  
Order Theory ◽  
Order Logic ◽  
Basic Theory ◽  
First Order Logic ◽  
Syntactic Category ◽  
First Order ◽  
Deductive Systems ◽  
First Order Theory ◽  
Categorical Semantics

This chapter provides the topos-theoretic background necessary for understanding the contents of the book; the presentation is self-contained and only assumes a basic familiarity with the language of category theory. The chapter begins by reviewing the basic theory of Grothendieck toposes, including the fundamental equivalence between geometric morphisms and flat functors. Then it presents the notion of first-order theory and the various deductive systems for fragments of first-order logic that will be considered in the course of the book, notably including that of geometric logic. Further, it discusses categorical semantics, i.e. the interpretation of first-order theories in categories possessing ‘enough’ structure. Lastly, the key concept of syntactic category of a first-order theory is reviewed; this notion will be used in Chapter 2 for constructing classifying toposes of geometric theories.

Polynomial rings and weak second-order logic

1985 ◽  
Vol 50 (4) ◽  
pp. 953-972 ◽  
Author(s):  
Anne Bauval
Keyword(s):  
Zero Divisor ◽  
Order Theory ◽  
Second Order ◽  
Order Logic ◽  
Polynomial Rings ◽  
Commutative Field ◽  
First Order ◽  
First Order Theory ◽  
Order Language ◽  
Second Order Logic

This article is a rewriting of my Ph.D. Thesis, supervised by Professor G. Sabbagh, and incorporates a suggestion from Professor B. Poizat. My main result can be crudely summarized (but see below for detailed statements) by the equality: first-order theory of F[Xi]i∈I = weak second-order theory of F.§I.1. Conventions. The letter F will always denote a commutative field, and I a nonempty set. A field or a ring (A; +, ·) will often be written A for short. We shall use symbols which are definable in all our models, and in the structure of natural numbers (N; +, ·):— the constant 0, defined by the formula Z(x): ∀y (x + y = y);— the constant 1, defined by the formula U(x): ∀y (x · y = y);— the operation ∹ x − y = z ↔ x = y + z;— the relation of division: x ∣ y ↔ ∃ z(x · z = y).A domain is a commutative ring with unity and without any zero divisor.By “… → …” we mean “… is definable in …, uniformly in any model M of L”.All our constructions will be uniform, unless otherwise mentioned.§I.2. Weak second-order models and languages. First of all, we have to define the models Pf(M), Sf(M), Sf′(M) and HF(M) associated to a model M = {A; ℐ) of a first-order language L [CK, pp. 18–20]. Let L1 be the extension of L obtained by adjunction of a second list of variables (denoted by capital letters), and of a membership symbol ∈. Pf(M) is the model (A, Pf(A); ℐ, ∈) of L1, (where Pf(A) is the set of finite subsets of A. Let L2 be the extension of L obtained by adjunction of a second list of variables, a membership symbol ∈, and a concatenation symbol ◠.

Gödel’s Proof Based on ω-Consistency

1992 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Predicate Symbol ◽  
Order Theory ◽  
Modus Ponens ◽  
Axiom System ◽  
Peano Arithmetic ◽  
Order Logic ◽  
First Order Logic ◽  
Underlying Assumption ◽  
First Order ◽  
First Order Theory

The proof that we have just given of the incompleteness of Peano Arithmetic was based on the underlying assumption that Peano Arithmetic is correct—i.e., that every sentence provable in P.A. is a true sentence. Gödel’s original incompleteness proof involved a much weaker assumption—that of ω-consistency to which we now turn. We consider an arbitrary axiom system S whose formulas are those of Peano Arithmetic, whose axioms include all those of Groups I and II (or alternatively, any set of axioms for first-order logic with identity such that all logically valid formulas are provable from them), and whose inference rules are modus ponens and generalization. (It is also possible to axiomatize first-order logic in such a way that modus ponens is the only inference rule—cf. Quine [1940].) In place of the axioms of Groups III and IV, however, we can take a completely arbitrary set of axioms. Such a system S is an example of what is termed a first-order theory, and we will consider several such theories other than Peano Arithmetic. (For the more general notion of a first-order theory, the key difference is that we do not necessarily start with + and × as the undefined function symbols, nor do we necessarily take ≤ as the undefined predicate symbol. Arbitrary function symbols and predicate symbols can be taken, however, as the undefined function and predicate symbols—cf. Tarski [1953] for details. However, the only theories (or “systems”, as we will call them) that we will have occasion to consider are those whose formulas are those of P.A.) S is called simply consistent (or just “consistent” for short) if no sentence is both provable and refutable in S.

scholarly journals Classifying Toposes for First Order Theories

1997 ◽  
Vol 4 (20) ◽  
Author(s):  
Carsten Butz ◽  
Peter T. Johnstone
Keyword(s):  
Order Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Conservative Extension ◽  
First Order ◽  
First Order Theory ◽  
Smallness Condition ◽  
Geometric Morphisms

By a classifying topos for a first-order theory T, we mean a topos<br />E such that, for any topos F, models of T in F correspond exactly to<br />open geometric morphisms F ! E. We show that not every (infinitary)<br />first-order theory has a classifying topos in this sense, but we<br />characterize those which do by an appropriate `smallness condition',<br />and we show that every Grothendieck topos arises as the classifying<br />topos of such a theory. We also show that every first-order theory<br /> has a conservative extension to one which possesses<br /> a classifying topos, and we obtain a Heyting-valued completeness<br /> theorem for infinitary first-order logic.

scholarly journals HYPER-REF: A General Model of Reference for First-Order Logic and First-Order Arithmetic

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Pablo Rivas-Robledo
Keyword(s):  
General Model ◽  
Heuristic Method ◽  
Order Theory ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
First Order Theory ◽  
Order Arithmetic ◽  
Self Reference ◽  
Do So

Abstract In this article I present HYPER-REF, a model to determine the referent of any given expression in First-Order Logic (FOL). I also explain how this model can be used to determine the referent of a first-order theory such as First-Order Arithmetic (FOA). By reference or referent I mean the non-empty set of objects that the syntactical terms of a well-formed formula (wff) pick out given a particular interpretation of the language. To do so, I will first draw on previous work to make explicit the notion of reference and its hyperintensional features. Then I present HYPER-REF and offer a heuristic method for determining the reference of any formula. Then I discuss some of the benefits and most salient features of HYPER-REF, including some remarks on the nature of self-reference in formal languages.

scholarly journals Second Order Logic or Set Theory?

2012 ◽  
Vol 18 (1) ◽  
pp. 91-121 ◽  
Author(s):  
Jouko Väänänen
Keyword(s):  
Set Theory ◽  
Order Theory ◽  
Second Order ◽  
Order Logic ◽  
First Order ◽  
First Order Theory ◽  
Standard Models ◽  
The Right ◽  
Second Order Logic ◽  
Proper Context

AbstractWe try to answer the question which is the “right” foundation of mathematics, second order logic or set theory. Since the former is usually thought of as a formal language and the latter as a first order theory, we have to rephrase the question. We formulate what we call the second order view and a competing set theory view, and then discuss the merits of both views. On the surface these two views seem to be in manifest conflict with each other. However, our conclusion is that it is very difficult to see any real difference between the two. We analyze a phenomenonwe call internal categoricity which extends the familiar categoricity results of second order logic to Henkin models and show that set theory enjoys the same kind of internal categoricity. Thus the existence of non-standard models, which is usually taken as a property of first order set theory, and categoricity, which is usually taken as a property of second order axiomatizations, can coherently coexist when put into their proper context. We also take a fresh look at complete second order axiomatizations and give a hierarchy result for second order characterizable structures. Finally we consider the problem of existence in mathematics from both points of view and find that second order logic depends on what we call large domain assumptions, which come quite close to the meaning of the axioms of set theory.

Some contributions to definability theory for languages with generalized quantifiers

1982 ◽  
Vol 47 (3) ◽  
pp. 572-586
Author(s):  
John T. Baldwin ◽  
Douglas E. Miller
Keyword(s):  
Order Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Natural Class ◽  
First Order ◽  
First Order Theory ◽  
First Results ◽  
Countably Compact ◽  
Compact Extension ◽  
Almost All

One of the first results in model theory [12] asserts that a first-order sentence is preserved in extensions if and only if it is equivalent to an existential sentence.In the first section of this paper, we analyze a natural program for extending this result to a class of languages extending first-order logic, notably including L(Q) and L(aa), respectively the languages with the quantifiers “there exist un-countably many” and “for almost all countable subsets”.In the second section we answer a question of Bruce [3] by showing that this program cannot resolve the question for L(Q). We also consider whether the natural class of “generalized Σ-sentences” in L(Q) characterizes the class of sentences preserved in extensions, refuting the relativized version but leaving the unrestricted question open.In the third section we show that the analogous class of L(aa)-sentences preserved in extensions does not include (up to elementary equivalence) all such sentences. This particular candidate class was nominated, rather tentatively, by Bruce [3].In the fourth section we show that under rather general conditions, if L is a countably compact extension of first-order logic and T is an ℵ1-categorical first-order theory, then L is trivial relative to T.

scholarly journals Generalized prime models

1971 ◽  
Vol 36 (4) ◽  
pp. 593-606 ◽  
Author(s):  
Robert Fittler
Keyword(s):  
Order Theory ◽  
Complete Theory ◽  
Prime Model ◽  
First Order ◽  
First Order Theory ◽  
General Conditions

A prime model O of some complete theory T is a model which can be elementarily imbedded into any model of T (cf. Vaught [7, Introduction]). We are going to replace the assumption that T is complete and that the maps between the models of T are elementary imbeddings (elementary extensions) by more general conditions. T will always be a first order theory with identity and may have function symbols. The language L(T) of T will be denumerable. The maps between models will be so called F-maps, i.e. maps which preserve a certain set F of formulas of L(T) (cf. I.1, 2). Roughly speaking a generalized prime model of T is a denumerable model O which permits an F-map O→M into any model M of T. Furthermore O has to be “generated” by formulas which belong to a certain subset G of F.

Nonarithmetical ℵ0-categorical theories with recursive models

1994 ◽  
Vol 59 (1) ◽  
pp. 106-112 ◽  
Author(s):  
Julia F. Knight
Keyword(s):  
Order Theory ◽  
Complete Theory ◽  
First Order ◽  
Recursive Model ◽  
Recursive Models ◽  
First Order Theory ◽  
Free Variables

In what follows, L is a recursive language. The structures to be considered are L-structures with universe named by constants from ω. A structure is recursive A if the open diagram D() is recursive. Lerman and Schmerl [L-S] proved the following result.Let T be an ℵ0-categorical elementary first-order theory. Suppose that for all n, , and T is arithmetical. Then T has a recursive model.The aim of this paper is to extend Theorem 0.1. Stating the extension requires some terminology. Consider finitary formulas with symbols from L and sometimes extra constants from ω. For each n ∈ ω, the Σn and Πn formulas are as usual. Then Bnformulas are Boolean combinations of Σn formulas. For an L-structure , Dn() denotes the set of Bn sentences in the complete diagram Dc(). A complete Σn theory is a maximal consistent set of ΣnL-sentences. We may write φ(x), or Γ(x), to indicate that the free variables of the formula φ, or the set Γ, are among those in x. A complete Bn type for x is a maximal consistent set Γ(x) of Bn formulas with just the free variables x.If T is ℵ0-categorical, then for each x only finitely many complete types Γ(x) are consistent with T. While Lerman and Schmerl stated their result just for ℵ0-categorical theories, essentially the same proof yields the following.Theorem 0.2. Let T be a consistent, complete theory such that for all n andx, only finitely many complete Bn types Γ(x) are consistent with T.

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