Piecewise Definable Theoretical Concepts

Studia Logica - An International Journal for Symbolic Logic - SLOGICA, 1994

The paper studies two formal schemes related to w-completeness. Let S be a suitable formal theory containing primitive recursive arithmetic and let T be a formal extension of S. Denoted by (a), (b) and (c), respectively, are the fonowing three propositions (where a(x) is a formula with the only free variable x): (a) (for any n) (F-T a(fi)), (b) b'T VxPrT(-C~(k)-) and (c) bT Vxa(x) (the notational conventions are those of Smoryfiski [3]). The aim of this paper is to examine the meaning of the schemes which result from the formalizations, over the base theory S, of the implications (b) =~ (c) and (a) =~ (b), where ot ranges over all formulae. The analysis yields two results over S : 1. the schema corresponding to (b) =~ (c) is equivalent to ~ConsT and 2. the schema corresponding to (a) =~ (b) is not consistent with 1-CONT. The former result follows from a simple adaptation of the w-incompleteness proof; the second is new and is based on a particular application of the diagonalization lemma. 1In particular, I have borrowed the following from Smoryfiski: the definitions of the predicates ProvT and PrT; the substitution function (restricted to numerals) within the provability predicate PrT(-C~(k)-); the symbols of the derivability conditions D1, D2 and D3 of Gbdel's theorems.

Annals of Mathematical Logic, 1970

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1979

2014

This is $a$ (non-exhaustive) collection of results addressing the question “If $A$ is such that $P(A)$ , does there exists a $B$ such that $Q(B)$ and $B$ is definable from $A?$”, for various properties $P(x),$ $Q(x)$ , as well as closely related questions. The focus is on classical combinatorial properties at the level of $H(\omega_{2})$ .

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1988

Journal of Computer and System Sciences, 1978

The equivalence between programs is an essential concept in the mathematical theory of computation and programming. That a program is correct, that a program transformation (as considered by Burstall and Darlington [6]) p reserves the computed function can be formulated in terms of equivalence between programs. But this relation is very difficult to study for well-known theoretical reasons, Program schemes have been introduced to overcome these difficulties as much as possible. A program P is divided into a program scheme $ (i.e., a program where the domain of computation and the base functions are left unspecified and an interpretation I (i.e., the specification of a domain D, and a function fi = Dlk-+ D, for each k-ary base function symbol). We denote by+, the function computed by $ under I, i.e., the function computed by the program P. We consider pure-LISP-like recursive program schemes without assignments. The conditional if. .. then. .. else. .., usually considered as a piece of control structure, for instance in [l], can be considered as a base function. Hence, in the corresponding schemes, it will be replaced by the 3-adic function symbol h(...,. .. .. ..). (Formal definitions are given in Section 2.) Since the function symbols can be interpreted by arbitrary functions of correct arity the corresponding equivalence relation on schemes, namely (b = $' iff $I = +; for every interpretation I, is very restrictive and does not help very much for the study of interesting equivalence between real programs. (See example 2.2 below.) In order to get more concrete results, we use the notion of a class of interpretations V; the associated equivalence between program schemes is then d =ud' iff 4, = 4; for every I E V.

2009

This paper introduces theories for arithmetical quasi-inductive definitions (Burgess, 1986) as it has been done for first-order monotone and nonmonotone inductive ones. After displaying the basic axiomatic framework, we provide some initial result in the proof theoretic bounds line of research (the upper one being given in terms of a theory of sets extending Kripke–Platek set theory).(Received May 04 2009)

JOHN CORCORAN. 1980. A note concerning definitional equivalence, History and Philosophy of Logic 1: 231–34. MR83j:01002. P R This paper defines certain purely syntactic homogeneous relations between formal, uninterpreted theories based on different formal languages that are sublanguages of a more extensive language. The fact that a theory is uninterpreted does not prevent it from having interpretations nor does it preclude the theory from being presented as the set of true sentences of a given concrete interpretation. The defined relations are called ‘interpretability’, ‘mutual interpretability’, and ‘definitional equivalence’. Let T0s be the set of sentences true in the standard interpretation of the second-order language of arithmetic based on primitives ‘0’ and ‘s’, for the number zero and the one-place operation of successor. Thus the constant terms of T1 are the strings ‘0’, ‘s0’, ‘ss0’, etc. Under the standard interpretation the universe of discourse of T1 is the set of natural numbers: 0, 1, 2, 3, etc. Let T0s^—note bold font—be the set of sentences true in the standard interpretation of the second-order language of two-character string theory based on primitives ‘0’, ‘s’, and ‘^’, for the zero numeral, the successor symbol, and the two-place operation of concatenation. Thus the constant terms of T0s^ are ‘0’, ‘s’, ‘(s^0)’, (0^ s)’ ‘(0^(s^0))’, etc. Under the standard interpretation the universe of discourse of T0s^ is the set of strings composed of the two characters ‘0’and ‘s’: the two strings of length one ‘0’, ‘s’, the four strings of length two ‘00’, ‘0s’, ‘s0’, ‘ss’, the eight strings of length three ‘000’, ‘00s’, ‘0s0’, ‘s00’ ‘0ss’, ‘s0s’, ‘ss0’, ‘sss’, the 16 strings of length 4, etc. As said in this paper, the reference Corcoran et al.1974 implies that T0s is definitional equivalent to T0s^. This does not mean that numbers are definable in terms of character strings. This does not mean that the successor operation is numbers definable in terms of character strings and concatenation. A fortiori, definitional equivalence of two theories implies nothing as to whether the ontological status of objects used to interpret one is prior to or dependent on the whether the ontological status of objects used to interpret the other. Admittedly, the terms ‘interpretability’, ‘mutual interpretability’, and ‘definitional equivalence’ may suggest otherwise to people who fail to notice that the three expressions refer to relations between uninterpreted theories. Perhaps, these and related issues concerning primitive concepts and primitive objects, as opposed to primitive symbols, can be discussed in the session.

2012

A downwards linear order is well-founded if and only if all its components are. In his study of definability [D], Doets ran into the question whether a similar invariance holds for definable well-foundedness. This question — the direction from right to left is the harder part — is settled below, in some additional generality. Moreover, all the difficult words of this introduction are explained there. 1. A definability theorem For any set X, let X * be the set of finite sequences of elements of X. Let A be a structure, fixed for this section, with universe A, for a first order language L. Let us assume for the sake of simplicity that all symbols of L are relation symbols. (We shall reconsider this assumption below.) Let B be a component of A: a subset of A with the property that for every symbol R of L, R A, the relation over A that is the interpretation of R, is contained in B * ∪ (A – B) *. I write B to refer to the substructure of A with universe B. I shall call the substructure a...

Publications de l'Institut Math?matique (Belgrade)

The concept of definition is usually not covered in mathematical logic textbooks. The definability of classes of structures is dealt with in model theory but the definability of concepts within a given structure is not. Our aim is to deal with these kind of definitions. We also address some of the implications for teaching and learning mathematics.