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.

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.

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.

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 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.

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.

Completeness in the theory of properties, relations, and propositions

1983 ◽  
Vol 48 (2) ◽  
pp. 415-426 ◽  
Author(s):  
George Bealer
Keyword(s):  
Possible Worlds ◽  
Order Theory ◽  
Order Logic ◽  
Present Approach ◽  
Intensional Logic ◽  
Comprehensive Treatment ◽  
First Order ◽  
First Order Theory ◽  
Technical Details ◽  
Logical Data

Higher-order theories of properties, relations, and propositions (PRPs) are known to be essentially incomplete relative to their standard notions of validity. There is, however, a first-order theory of PRPs that results when standard first-order logic is supplemented with an operation of intensional abstraction. It turns out that this first-order theory of PRPs is provably complete with respect to its standard notions of validity. The construction involves the development of a new algebraic semantic method. Unlike most other methods used in contemporary intensional logic, this method does not appeal to possible worlds as a heuristic; the heuristic used is that of PRPs taken as primitive entities. This is important, for even though the possible-worlds approach is useful in treating modal logic, it seems to be of little help in treating the logic for psychological matters. The present approach, by contrast, appears to make a step in the direction of a satisfactory treatment of both modal and intentional logic. For, by taking PRPs as primitive entities, we remain free to tailor the statement of their identity conditions so that it agrees with the logical data—modal, psychological, etc. In this way, the present approach suggests a strategy for developing a comprehensive treatment of intensional logic.In [1] and [2] I explore this prospect philosophically. The purpose of the present paper is to lay out the technical details of the approach and to present the completeness results.

scholarly journals HOL-λσ: an intentional first-order expression of higher-order logic

2001 ◽  
Vol 11 (1) ◽  
pp. 21-45 ◽  
Author(s):  
GILLES DOWEK ◽  
THERESE HARDIN ◽  
CLAUDE KIRCHNER
Keyword(s):  
Order Theory ◽  
Higher Order ◽  
Search Method ◽  
Order Logic ◽  
Automated Deduction ◽  
Higher Order Logic ◽  
First Order ◽  
Proof Search ◽  
First Order Theory ◽  
Explicit Substitutions

We give a first-order presentation of higher-order logic based on explicit substitutions. This presentation is intentionally equivalent to the usual presentation of higher-order logic based on λ-calculus, that is, a proposition can be proved without the extensionality axioms in one theory if and only if it can be in the other. We show that the Extended Narrowing and Resolution first-order proof-search method can be applied to this theory. In this way we get a step-by-step simulation of higher-order resolution. Hence, expressing higher-order logic as a first-order theory and applying a first-order proof search method is a relevant alternative to a direct implementation. In particular, the well-studied improvements of proof search for first-order logic could be reused at no cost for higher-order automated deduction. Moreover, as we stay in a first-order setting, extensions, such as equational higher-order resolution, may be easier to handle.

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 ◠.

Finite Variable Logics in Descriptive Complexity Theory

1998 ◽  
Vol 4 (4) ◽  
pp. 345-398 ◽  
Author(s):  
Martin Grohe
Keyword(s):  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Complexity Classes ◽  
Finite Model ◽  
Polynomial Space ◽  
Finite Model Theory ◽  
First Order ◽  
First Order Theory ◽  
Finite Structures

Throughout the development of finite model theory, the fragments of first-order logic with only finitely many variables have played a central role. This survey gives an introduction to the theory of finite variable logics and reports on recent progress in the area.For each k ≥ 1 we let Lk be the fragment of first-order logic consisting of all formulas with at most k (free or bound) variables. The logics Lk are the simplest finite-variable logics. Later, we are going to consider infinitary variants and extensions by so-called counting quantifiers.Finite variable logics have mostly been studied on finite structures. Like the whole area of finite model theory, they have interesting model theoretic, complexity theoretic, and combinatorial aspects. For finite structures, first-order logic is often too expressive, since each finite structure can be characterized up to isomorphism by a single first-order sentence, and each class of finite structures that is closed under isomorphism can be characterized by a first-order theory. The finite variable fragments seem to be promising candidates with the right balance between expressive power and weakness for a model theory of finite structures. This may have motivated Poizat [67] to collect some basic model theoretic properties of the Lk. Around the same time Immerman [45] showed that important complexity classes such as polynomial time (PTIME) or polynomial space (PSPACE) can be characterized as collections of all classes of (ordered) finite structures definable by uniform sequences of first-order formulas with a fixed number of variables and varying quantifier-depth.

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