Partially ordered interpretations

1977 ◽  
Vol 42 (1) ◽  
pp. 83-93
Author(s):  
Nobuyoshi Motohashi
Keyword(s):  
Basic Result ◽  
Predicate Symbol ◽  
Order Theory ◽  
Free Variable ◽  
Consistent Theory ◽  
First Order ◽  
Partially Ordered ◽  
First Order Theory ◽  
Order Language ◽  
Special Case

In this paper, we shall define the “partially ordered interpretation” of a first order theory in another first order theory and state some recent results. Although an exact definition will be given in §4 below, we now give a brief outline. First of all, let us recall the “interpretations” defined by A. Tarski et al. in [17] and the “parametrical interpretations” defined by P. Hájek in [6], [7] and U. Felgner in [3]. Since “interpretations” can be considered as a special case of “parametrical interpretations”, we consider only the latter type of “interpretations”. A parametrical interpretation I of a first order language L in a consistent theory T′ (formulated in another first order language L′) consists of the following formulas:(i) a unary formula C(p) (i.e. a formula with one designated free variable p), which is used to denote the range of parameters,(ii) a binary formula U(p, x), which is intended to denote the pth universe for each parameter p,(iii) an (n + 1)-ary formula Fp(p, x1 …, xn) for each n-ary predicate symbol P in L,such that the formulas (∃p)C(p) and (∀p)(C(p)→(∃x)U(p, x)) are provable in T". Then, given a formula A in L and a parameter p, we define the interpretation Ip (A ) of A by I at p to be the formula which is obtained from A by replacing every atomic subformula P(*, …, *) in A by Fp(p, *,…,*), and relativizing every occurrence of quantifiers in A by U(p, * ). A sentence A in L is said to be I-provable in T′ if the sentence (∀p) (C(p)→ Ip(A)) is provable in T′. Then, it is obvious that every provable sentence in L is I-provable in T′. This is a basic result of “parametrical interpretations” and is used to prove the “consistency” of a theory T in L by showing that every axiom of T is I-provable in T′ when I is said to be a parametrical interpretation of T in T′. As is shown above, the word “interpretation” is used in the following three senses: interpretations of languages, interpretations of formulas and interpretations of theories. So, in this introduction we let the word “interpretation” denote “interpretation of languages”, for short.

Decidable properties of finite sets of equations in trivial languages

1984 ◽  
Vol 49 (4) ◽  
pp. 1333-1338
Author(s):  
Cornelia Kalfa
Keyword(s):  
Equational Theory ◽  
Order Theory ◽  
Operation Symbol ◽  
First Order ◽  
First Order Theory ◽  
Order Language ◽  
Finite Set ◽  
Joint Embedding ◽  
Algebraic Language ◽  
Joint Embedding Property

In [4] I proved that in any nontrivial algebraic language there are no algorithms which enable us to decide whether a given finite set of equations Σ has each of the following properties except P2 (for which the problem is open):P0(Σ) = the equational theory of Σ is equationally complete.P1(Σ) = the first-order theory of Σ is complete.P2(Σ) = the first-order theory of Σ is model-complete.P3(Σ) = the first-order theory of the infinite models of Σ is complete.P4(Σ) = the first-order theory of the infinite models of Σ is model-complete.P5(Σ) = Σ has the joint embedding property.In this paper I prove that, in any finite trivial algebraic language, such algorithms exist for all the above Pi's. I make use of Ehrenfeucht's result [2]: The first-order theory generated by the logical axioms of any trivial algebraic language is decidable. The results proved here are part of my Ph.D. thesis [3]. I thank Wilfrid Hodges, who supervised it.Throughout the paper is a finite trivial algebraic language, i.e. a first-order language with equality, with one operation symbol f of rank 1 and at most finitely many constant symbols.

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 NIP FOR THE ASYMPTOTIC COUPLE OF THE FIELD OF LOGARITHMIC TRANSSERIES

2017 ◽  
Vol 82 (1) ◽  
pp. 35-61 ◽  
Author(s):  
ALLEN GEHRET
Keyword(s):  
Order Theory ◽  
Exchange Property ◽  
Independence Property ◽  
First Order ◽  
Valued Field ◽  
First Order Theory ◽  
Order Language ◽  
Complete Survey ◽  
Image Position

AbstractThe derivation on the differential-valued field Tlog of logarithmic transseries induces on its value group ${{\rm{\Gamma }}_{{\rm{log}}}}$ a certain map ψ. The structure ${\rm{\Gamma }} = \left( {{{\rm{\Gamma }}_{{\rm{log}}}},\psi } \right)$ is a divisible asymptotic couple. In [7] we began a study of the first-order theory of $\left( {{{\rm{\Gamma }}_{{\rm{log}}}},\psi } \right)$ where, among other things, we proved that the theory $T_{{\rm{log}}} = Th\left( {{\rm{\Gamma }}_{{\rm{log}}} ,\psi } \right)$ has a universal axiomatization, is model complete and admits elimination of quantifiers (QE) in a natural first-order language. In that paper we posed the question whether Tlog has NIP (i.e., the Non-Independence Property). In this paper, we answer that question in the affirmative: Tlog does have NIP. Our method of proof relies on a complete survey of the 1-types of Tlog, which, in the presence of QE, is equivalent to a characterization of all simple extensions ${\rm{\Gamma }}\left\langle \alpha \right\rangle$ of ${\rm{\Gamma }}$. We also show that Tlog does not have the Steinitz exchange property and we weigh in on the relationship between models of Tlog and the so-called precontraction groups of [9].

Some consequences of the axiom of power-set

1965 ◽  
Vol 30 (3) ◽  
pp. 293-294 ◽  
Author(s):  
Alexander Abian ◽  
Samuel Lamacchia
Keyword(s):  
Predicate Symbol ◽  
Order Theory ◽  
Finite Model ◽  
First Order ◽  
First Order Theory ◽  
Power Set ◽  
Binary Predicate ◽  
Definition Of

In this paper we prove:Theorem 1. Any finite model of the axiom of power-set also satisfies the axioms of extensionality, sum-set and choice.Clearly, it will follow from (2) below that in a finite model the axiom of power-set is satisfied if and only if every set is a power-set. Thus, Theorem 1 follows immediately from Theorem 2 below, where by a theory of sets we mean a first-order theory without identity and with only one binary predicate symbol ∈.Theorem 2. If in a theory of sets every set is a power-set and if the axiom of power-set is valid, then the axioms of extensionality, sum-set and choice are valid.The proof of Theorem 2 will follow from the lemmas which we establish below.We mean by x = y that x and y have the same elements. We denote a power-set of x by P(x) when it exists; similarly, we denote a sum-set of x by Ux.Clearly, in every theory of sets we have:(1) (x ⊂ y) ↔ (P(x) ⊂ P(y)),(2) (x = y) ↔ (P(x) = P(y)),(3) (x = y) → ((x ∈ P(z)) → (y ∈ P(z))),(4) ⋃P(x) = x.In view of (2), (3) and the definition of equality, we have:Lemma 1. If in a theory of sets every set is a power-set, then equal sets are elements of the same sets.We have also, in view of (4):Lemma 2. If in a theory of sets every set is a power-set, then every set has a sum-set.

The incompleteness theorems

2004 ◽  
Author(s):  
Shawn Hedman
Keyword(s):  
Mathematical Reasoning ◽  
Order Theory ◽  
Point Of View ◽  
Incompleteness Theorem ◽  
Bertrand Russell ◽  
First Order ◽  
First Order Theory ◽  
Incompleteness Theorems ◽  
Special Case ◽  
Gödel’S Incompleteness Theorems

In this chapter we prove that the structure N = (ℕ|+, · , 1) has a first-order theory that is undecidable. This is a special case of Gödel’s First Incompleteness theorem. This theorem implies that any theory (not necessarily first-order) that describes elementary arithmetic on the natural numbers is necessarily undecidable. So there is no algorithm to determine whether or not a given sentence is true in the structure N. As we shall show, the existence of such an algorithm leads to a contradiction. Gödel’s Second Incompleteness theorem states that any decidable theory (not necessarily first-order) that can express elementary arithmetic cannot prove its own consistency. We shall make this idea precise and discuss the Second Incompleteness theorem in Section 8.5. Gödel’s First Incompleteness theorem is proved in Section 8.3. Although they are purely mathematical results, Gödel’s Incompleteness theorems have had undeniable philosophical implications. Gödel’s theorems dispelled commonly held misconceptions regarding the nature of mathematics. A century ago, some of the most prominent mathematicians and logicians viewed mathematics as a branch of logic instead of the other way around. It was thought that mathematics could be completely formalized. It was believed that mathematical reasoning could, at least in principle, be mechanized. Alfred North Whitehead and Bertrand Russell envisioned a single system that could be used to derive and enumerate all mathematical truths. In their three-volume Principia Mathematica, Russell and Whitehead rigorously define a system and use it to derive numerous known statements of mathematics. Gödel’s theorems imply that any such system is doomed to be incomplete. If the system is consistent (which cannot be proved within the system by Gödel’s Second theorem), then there necessarily exist true statements formulated within the system that the system cannot prove (by Gödel’s First theorem). This explains why the name “incompleteness” is attributed to these theorems and why the title of Gödel’s 1931 paper translates (from the original German) to “On Formally Undecidable Propositions of Principia Mathematica and Related Systems” (translated versions appear in both [13] and [14]). Depending on one’s point of view, it may or may not be surprising that there is no algorithm to determine whether or not a given sentence is true in N.

scholarly journals An Invitation to Model-Theoretic Galois Theory

2010 ◽  
Vol 16 (2) ◽  
pp. 261-269 ◽  
Author(s):  
Alice Medvedev ◽  
Ramin Takloo-Bighash
Keyword(s):  
Galois Group ◽  
Galois Theory ◽  
Order Theory ◽  
Closed Field ◽  
Finite Sets ◽  
Algebraically Closed Field ◽  
First Order ◽  
First Order Theory ◽  
Special Case ◽  
Large Model

AbstractWe carry out some of Galois' work in the setting of an arbitrary first-order theory T. We replace the ambient algebraically closed field by a large model M of T, replace fields by definably closed subsets of M, assume that T codes finite sets, and obtain the fundamental duality of Galois theory matching subgroups of the Galois group of L over F with intermediate extensions F ≤ K ≤ L. This exposition of a special case of [10] has the advantage of requiring almost no background beyond familiarity with fields, polynomials, first-order formulae, and automorphisms.

Locally finite theories

1986 ◽  
Vol 51 (1) ◽  
pp. 59-62 ◽  
Author(s):  
Jan Mycielski
Keyword(s):  
Main Idea ◽  
Order Theory ◽  
Interpolation Theorem ◽  
Point Of View ◽  
Finite Model ◽  
Consistent Theory ◽  
Locally Finite ◽  
First Order ◽  
First Order Theory ◽  
Finite Theory

We say that a first order theoryTislocally finiteif every finite part ofThas a finite model. It is the purpose of this paper to construct in a uniform way for any consistent theoryTa locally finite theory FIN(T) which is syntactically (in a sense) isomorphic toT.Our construction draws upon the main idea of Paris and Harrington [6] (I have been influenced by some unpublished notes of Silver [7] on this subject) and generalizes the syntactic aspect of their result from arithmetic to arbitrary theories. (Our proof is syntactic, and it is simpler than the proofs of [5], [6] and [7]. This reminds me of the simple syntactic proofs of several variants of the Craig-Lyndon interpolation theorem, which seem more natural than the semantic proofs.)The first mathematically strong locally finite theory, called FIN, was defined in [1] (see also [2]). Now we get much stronger ones, e.g. FIN(ZF).From a physicalistic point of view the theorems of ZF and their FIN(ZF)-counterparts may have the same meaning. Therefore FIN(ZF) is a solution of Hilbert's second problem. It eliminates ideal (infinite) objects from the proofs of properties of concrete (finite) objects.In [4] we will demonstrate that one can develop a direct finitistic intuition that FIN(ZF) is locally finite. We will also prove a variant of Gödel's second incompleteness theorem for the theory FIN and for all its primitively recursively axiomatizable consistent extensions.The results of this paper were announced in [3].

Categorical Logic and Model Theory

2018 ◽  
Author(s):  
John L. Bell
Keyword(s):  
Geometric Theory ◽  
Order Theory ◽  
Generic Model ◽  
Higher Order Logic ◽  
First Order ◽  
Categorical Logic ◽  
Deductive Systems ◽  
First Order Theory ◽  
Order Language ◽  
The 1960S

The chapter begins with an introduction describing the development of categorical logic from the 1960s. The next section, `Categories and Deductive Systems’, describes the relationship between categories and propositional logic, while the ensuing section, `Functorial Semantics’, is devoted to Lawvere’s provision of the first-order theory of models with a categorical formulation. In the section `Local Set Theories and Toposes’ the categorical counterparts—toposes—to higher-order logic are introduced, along with their associated theories—local set theories. In the section `Models of First-Order Languages in Categories’ the idea of an interpretation of a many-sorted first-order language is introduced, along with the concept of generic model of a theory formulated in such a language. The chapter concludes with the section `Models in Toposes’, wherein is introduced the concept of a first-order geometric theory and its associated classifying topos containing a generic model of the theory.

scholarly journals A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 119
Author(s):  
Marcoen J. T. F. Cabbolet
Keyword(s):  
Set Theory ◽  
Category Theory ◽  
Present Theory ◽  
Order Theory ◽  
Countable Model ◽  
First Order ◽  
First Order Theory ◽  
Order Language ◽  
Skolem Theorem ◽  
Axiomatic Set Theory

It is well known that Zermelo-Fraenkel Set Theory (ZF), despite its usefulness as a foundational theory for mathematics, has two unwanted features: it cannot be written down explicitly due to its infinitely many axioms, and it has a countable model due to the Löwenheim–Skolem theorem. This paper presents the axioms one has to accept to get rid of these two features. For that matter, some twenty axioms are formulated in a non-classical first-order language with countably many constants: to this collection of axioms is associated a universe of discourse consisting of a class of objects, each of which is a set, and a class of arrows, each of which is a function. The axioms of ZF are derived from this finite axiom schema, and it is shown that it does not have a countable model—if it has a model at all, that is. Furthermore, the axioms of category theory are proven to hold: the present universe may therefore serve as an ontological basis for category theory. However, it has not been investigated whether any of the soundness and completeness properties hold for the present theory: the inevitable conclusion is therefore that only further research can establish whether the present results indeed constitute an advancement in the foundations of mathematics.

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