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isomorphic toT.Our construction draws upon the main idea of Paris and Harrington [6] (I have been influenced by some unpublished notes of Silver [7] on this subject) and generalizes the syntactic aspect of their result from arithmetic to arbitrary theories.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].

On models of the elementary theory of (Z, +, 1)

1990 ◽  
Vol 55 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Mark Nadel ◽  
Jonathan Stavi
Keyword(s):  
Stability Theory ◽  
Elementary Theory ◽  
Additive Group ◽  
Order Theory ◽  
Point Of View ◽  
Real Numbers ◽  
First Order ◽  
First Order Theory ◽  
Distinguished Element ◽  
Unbounded Subset

Let T1 be the complete first-order theory of the additive group of the integers with 1 as distinguished element (in symbols, T1 = Th(Z, +, 1)). In this paper we prove that all models of T1 are ℵ0-homogeneous (§2), classify them (and lists of elements in them) up to isomorphism or L∞κ-equivalence (§§3 and 4) and show that they may be as complex as arbitrary sets of real numbers from the point of view of admissible set theory (§5). The results of §§2 and 5 together show that while the Scott heights of all models of T1 are ≤ ω (by ℵ0-homogeneity) their HYP-heights form an unbounded subset of the cardinal .In addition to providing this unusual example of the relation between Scott heights and HYP-heights, the theory T1 has served (using the homogeneity results of §2) as an example for certain combinations of properties that people had looked for in stability theory (see end of §4). In §6 it is shown that not all models of T = Th(Z, +) are ℵ0-homogeneous, so that the availability of the constant for 1 is essential for the result of §2.The two main results of this paper (2.2 and essentially Theorem 5.3) were obtained in the summer of 1979. Later we learnt from Victor Harnik and Julia Knight that T1 is of some interest for stability theory, and were encouraged to write up our proofs.During 1982/3 we improved the proofs and added some results.

SOME PROPERTIES OF FINITELY DECIDABLE VARIETIES

1992 ◽  
Vol 02 (01) ◽  
pp. 89-101 ◽  
Author(s):  
MATTHEW A. VALERIOTE ◽  
ROSS WILLARD
Keyword(s):  
Order Theory ◽  
Subdirectly Irreducible ◽  
Locally Finite ◽  
First Order ◽  
Minimal Sets ◽  
First Order Theory ◽  
Finite Member ◽  
Transfer Principles

Let [Formula: see text] be a variety whose class of finite members has a decidable first-order theory. We prove that each finite member A of [Formula: see text] satisfies the (3, 1) and (3, 2) transfer principles, and that the minimal sets of prime quotients of type 2 or 3 in A must have empty tails. The first result has already been used by J. Jeong [9] in characterizing the finite subdirectly irreducible members of [Formula: see text] with nonabelian monolith. The second result implies that if [Formula: see text] is also locally finite and omits type 1, then [Formula: see text] is congruence modular.

LOGICAL ASPECTS OF CAYLEY-GRAPHS: THE MONOID CASE

2006 ◽  
Vol 16 (02) ◽  
pp. 307-340 ◽  
Author(s):  
DIETRICH KUSKE ◽  
MARKUS LOHREY
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Order Theory ◽  
Second Order ◽  
Point Of View ◽  
Direct Products ◽  
Graph Products ◽  
Arbitrary Graph ◽  
First Order ◽  
First Order Theory

Cayley-graphs of monoids are investigated under a logical point of view. It is shown that the class of monoids, for which the Cayley-graph has a decidable monadic second-order theory, is closed under free products. This result is derived from a result of Walukiewicz, stating that the decidability of monadic second-order theories is preserved under tree-like unfoldings. Concerning first-order logic, it is shown that the class of monoids, for which the Cayley-graph has a decidable first-order theory, is closed under arbitrary graph products, which generalize both, free and direct products. For the proof of this result, tree-like unfoldings are generalized to so-called factorized unfoldings. It is shown that the decidability of the first-order theory of a structure is preserved by factorized unfoldings. Several additional results concerning factorized unfoldings are shown.

Prime e.c. commutative rings in characteristic n ≥ 2

1999 ◽  
Vol 64 (2) ◽  
pp. 629-633
Author(s):  
Dan Saracino
Keyword(s):  
Commutative Rings ◽  
Order Theory ◽  
Prime Model ◽  
Finite Model ◽  
First Order ◽  
First Order Theory ◽  
Building Models ◽  
Open Questions ◽  
A Chain ◽  
Mathematics Department

Let CR denote the first-order theory of commutative rings with unity, formulated in the language L = 〈 +, •, 0, 1〉. Virtually everything that is known about existentially complete (e.c.) models of CR is contained in Cherlin's paper [2], where it is shown, in particular, that the e.c. models are not first-order axiomatizable. The purpose of this note is to show that, in analogy with the case of fields, there exists a unique prime e.c. model of CR in each characteristic n > 2. As a consequence we settle Problem 8 in the list of open questions at the end of Hodges' book Building models by games ([5], p. 278).By a “prime” e.c. model of characteristic n ≥ 2 we mean one that embeds in every e.c. model of characteristic n. (The embedding is not always elementary, since [2] not all e.c. models of characteristic n are elementarily equivalent.) The prime model is characterized by the fact that it is the union of a chain of finite subrings each of which is an amalgamation base for CR. In §1 we describe the finite amalgamation bases for CR and show that every finite model embeds in a finite amalgamation base; in §2 we use this information to obtain prime e.c. models and answer Hodges' question.Our results on prime e.c. models were obtained some years ago, during the fall term of 1982, while the author was a visitor at Wesleyan University. The author wishes to take this opportunity to thank the mathematics department at Wesleyan for its hospitality during that visit, and subsequent ones.

A NEW FORMALIZATION OF FEFERMAN’S SYSTEM OF FUNCTIONS AND CLASSES AND ITS RELATION TO FREGE STRUCTURE

1995 ◽  
Vol 06 (03) ◽  
pp. 187-202 ◽  
Author(s):  
SUSUMU HAYASHI ◽  
SATOSHI KOBAYASHI
Keyword(s):  
Finite Number ◽  
Order Theory ◽  
Point Of View ◽  
The Other ◽  
Technical Result ◽  
Technical Point ◽  
First Order ◽  
First Order Theory

A new axiomatization of Feferman’s systems of functions and classes1,2 is given. The new axiomatization has a finite number of class constructors resembling the proposition constructors of Frege structure by Aczel.3 Aczel wrote “It appears that from the technical point of view the two approaches (Feferman’s system and Frege structure) run parallel to each other in the sense that any technical result for one approach can be reconstructed for the other”.3 By the aid of the new axiomatization, Aczel’s observation becomes so evident. It is now straightforward to give a mutual interpretation between our formulation and a first order theory of Frege structure, which improve results by Beeson in Ref. 4.

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.

An Example of a Complete First-Order Theory with All Models Algorithmically Trivial but Without Locally Finite Models

1982 ◽  
Vol 5 (3-4) ◽  
pp. 313-318
Author(s):  
Paweł Urzyczyn
Keyword(s):  
Order Theory ◽  
Complete Theory ◽  
Prime Model ◽  
Program Schema ◽  
Locally Finite ◽  
First Order ◽  
Finite Models ◽  
First Order Theory

We show an example of a first-order complete theory T, with no locally finite models and such that every program schema, total over a model of T, is strongly equivalent in that model to a loop-free schema. For this purpose we consider the notion of an algorithmically prime model, what enables us to formulate an analogue to Ryll-Nardzewski Theorem.

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 Note About a New Evaluation of the Direct Perturbations of the Planets on the Moon’s Motion

1981 ◽  
Vol 63 ◽  
pp. 265-266
Author(s):  
D. Standaert
Keyword(s):  
Order Theory ◽  
Second Order ◽  
Point Of View ◽  
Internal Point ◽  
First Order ◽  
Keplerian Orbits ◽  
First Order Theory ◽  
External Point ◽  
Constants Of Motion ◽  
The Mean

The aim of this paper is to present the principal features of a new evaluation of the direct perturbations of the planets on the Moon’s motion. Using the method already published in Celestial Mechanics (Standaert, 1980), we compute “a first-order theory” aiming at an accuracy of the order of the meter for all periodic terms of period less than 3 500 years.From an external point of view, we mean by that: a)keplerian orbits for the planets,b)the ELP-2000 solution of the Main Problem proposed by Mrs. Chapront (Chapront-Touzë, 1980),c)the first-order derivatives with respect to the constants of motion of the SALE theory of Henrard (Henrard, 1979).On the other hand, from an internal point of view, the computations include: d)the development in Legendre polynomials not only to the first-order in (a/a'), but also the following ones (up to the sixth-order for Venus, for example),e)the contributions of the second-order in the Lie triangle,f)second-order contributions coming from the corrections of the mean motions due to the planetary action.

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