On undecidable statements in enlarged systems of logic and the concept of truth

1939 ◽  
Vol 4 (3) ◽  
pp. 105-112 ◽  
Author(s):  
Alfred Tarski
Keyword(s):  
Set Theory ◽  
Modus Ponens ◽  
Logical System ◽  
Natural Numbers ◽  
System L ◽  
Concept Of Truth ◽  
Made In

It is my intention in this paper to add somewhat to the observations already made in my earlier publications on the existence of undecidable statements in systems of logic possessing rules of inference of a “non-finitary” (“non-constructive”) character (§§1–4).I also wish to emphasize once more the part played by the concept of truth in relation to problems of this nature (§§5–8).At the end of this paper I shall give a result which was not touched upon in my earlier publications. It seems to be of interest for the reason (among others) that it is an example of a result obtained by a fruitful combination of the method of constructing undecidable statements (due to K. Gödel) with the results obtained in the theory of truth.1. Let us consider a formalized logical system L. Without giving a detailed description of the system we shall assume that it possesses the usual “finitary” (“constructive”) rules of inference, such as the rule of substitution and the rule of detachment (modus ponens), and that among the laws of the system are included all the postulates of the calculus of statements, and finally that the laws of the system suffice for the construction of the arithmetic of natural numbers. Moreover, the system L may be based upon the theory of types and so be the result of some formalization of Principia mathematica. It may also be a system which is independent of any theory of types and resembles Zermelo's set theory.

Certain predicates defined by induction schemata

1953 ◽  
Vol 18 (1) ◽  
pp. 49-59 ◽  
Author(s):  
Hao Wang
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Formal System ◽  
General Notion ◽  
Constant Number ◽  
Natural Numbers ◽  
System L ◽  
Truth Definition ◽  
Consistency Proofs ◽  
Special Case

It is known that we can introduce in number theory (for example, the system Z of Hilbert-Bernays) by induction schemata certain predicates of natural numbers which cannot be expressed explicitly within the framework of number theory. The question arises how we can define these predicates in some richer system, without employing induction schemata. In this paper a general notion of definability by induction (relative to number theory), which seems to apply to all the known predicates of this kind, is introduced; and it is proved that in a system L1 which forms an extension of number theory all predicates which are definable by induction (hereafter to be abbreviated d.i.) according to the definition are explicitly expressible.In order to define such predicates and prove theorems answering to their induction schemata, we have to allow certain impredicative classes in L1. However, if we want merely to prove that for each constant number the special case of the induction schema for a predicate d.i. is provable, we do not have to assume the existence of impredicative classes. A certain weaker system L2, in which only predicative classes of natural numbers are allowed, is sufficient for the purpose. It is noted that a truth definition for number theory can be obtained in L2. Consistency proofs for number theory do not seem to be formalizable in L2, although they can, it is observed, be formalized in L1.In general, given any ordinary formal system (say Zermelo set theory), it is possible to define by induction schemata, in the same manner as in number theory, certain predicates which are not explicitly definable in the system. Here again, by extending the system in an analogous fashion, these predicates become expressible in the resulting system. The crucial predicate instrumental to obtaining a truth definition for a given system is taken as an example.

Transcendence of cardinals

1965 ◽  
Vol 30 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Gaisi Takeuti
Keyword(s):  
Set Theory ◽  
Natural Numbers ◽  
Recursive Functions ◽  
Power Set ◽  
Recursive Predicate

In this paper, by a function of ordinals we understand a function which is defined for all ordinals and each of whose value is an ordinal. In [7] (also cf. [8] or [9]) we defined recursive functions and predicates of ordinals, following Kleene's definition on natural numbers. A predicate will be called arithmetical, if it is obtained from a recursive predicate by prefixing a sequence of alternating quantifiers. A function will be called arithmetical, if its representing predicate is arithmetical.The cardinals are identified with those ordinals a which have larger power than all smaller ordinals than a. For any given ordinal a, we denote by the cardinal of a and by 2a the cardinal which is of the same power as the power set of a. Let χ be the function such that χ(a) is the least cardinal which is greater than a.Now there are functions of ordinals such that they are easily defined in set theory, but it seems impossible to define them as arithmetical ones; χ is such a function. If we define χ in making use of only the language on ordinals, it seems necessary to use the notion of all the functions from ordinals, e.g., as in [6].

scholarly journals Selectively Pseudocompact Groups without Infinite Separable Pseudocompact Subsets

Axioms ◽  
2018 ◽  
Vol 7 (4) ◽  
pp. 86 ◽  
Author(s):  
Dmitri Shakhmatov ◽  
Víctor Yañez
Keyword(s):  
Topological Group ◽  
Set Theory ◽  
Axiom Of Choice ◽  
Natural Numbers ◽  
Compactness Property ◽  
Free Ultrafilter ◽  
The Axiom Of Choice

We give a “naive” (i.e., using no additional set-theoretic assumptions beyond ZFC, the Zermelo-Fraenkel axioms of set theory augmented by the Axiom of Choice) example of a Boolean topological group G without infinite separable pseudocompact subsets having the following “selective” compactness property: For each free ultrafilter p on the set N of natural numbers and every sequence ( U n ) of non-empty open subsets of G, one can choose a point x n ∈ U n for all n ∈ N in such a way that the resulting sequence ( x n ) has a p-limit in G; that is, { n ∈ N : x n ∈ V } ∈ p for every neighbourhood V of x in G. In particular, G is selectively pseudocompact (strongly pseudocompact) but not selectively sequentially pseudocompact. This answers a question of Dorantes-Aldama and the first listed author. The group G above is not pseudo- ω -bounded either. Furthermore, we show that the free precompact Boolean group of a topological sum ⨁ i ∈ I X i , where each space X i is either maximal or discrete, contains no infinite separable pseudocompact subsets.

Frege's theorem in a constructive setting

1999 ◽  
Vol 64 (2) ◽  
pp. 486-488 ◽  
Author(s):  
John L. Bell
Keyword(s):  
Set Theory ◽  
Natural Numbers ◽  
The Family ◽  
Power Set ◽  
Law Of Excluded Middle ◽  
The Law ◽  
Image Position ◽  
Excluded Middle ◽  
Intuitionistic Set Theory

By Frege's Theorem is meant the result, implicit in Frege's Grundlagen, that, for any set E, if there exists a map υ from the power set of E to E satisfying the conditionthen E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in Section 3 of [1].) My purpose in this note is to strengthen this result in two directions: first, the premise will be weakened so as to require only that the map υ be defined on the family of (Kuratowski) finite subsets of the set E, and secondly, the argument will be constructive, i.e., will involve no use of the law of excluded middle. To be precise, we will prove, in constructive (or intuitionistic) set theory, the followingTheorem. Let υ be a map with domain a family of subsets of a set E to E satisfying the following conditions:(i) ø ϵdom(υ)(ii)∀U ϵdom(υ)∀x ϵ E − UU ∪ x ϵdom(υ)(iii)∀UV ϵdom(5) υ(U) = υ(V) ⇔ U ≈ V.Then we can define a subset N of E which is the domain of a model of Peano's axioms.

Inconsistency of uncountable infinite sets under ZFC framework

Kybernetes ◽  
2008 ◽  
Vol 37 (3/4) ◽  
pp. 453-457 ◽  
Author(s):  
Wujia Zhu ◽  
Yi Lin ◽  
Guoping Du ◽  
Ningsheng Gong
Keyword(s):  
Natural Number ◽  
Set Theory ◽  
Design Methodology ◽  
Natural Numbers ◽  
Real Numbers ◽  
Conceptual Approach ◽  
Content Type ◽  
Infinite Sets ◽  
Axiomatic Set Theory ◽  
First Time

PurposeThe purpose is to show that all uncountable infinite sets are self‐contradictory non‐sets.Design/methodology/approachA conceptual approach is taken in the paper.FindingsGiven the fact that the set N={x|n(x)} of all natural numbers, where n(x)=df “x is a natural number” is a self‐contradicting non‐set in this paper, the authors prove that in the framework of modern axiomatic set theory ZFC, various uncountable infinite sets are either non‐existent or self‐contradicting non‐sets. Therefore, it can be astonishingly concluded that in both the naive set theory or the modern axiomatic set theory, if any of the actual infinite sets exists, it must be a self‐contradicting non‐set.Originality/valueThe first time in history, it is shown that such convenient notion as the set of all real numbers needs to be reconsidered.

A theory of properties

1987 ◽  
Vol 52 (2) ◽  
pp. 455-472 ◽  
Author(s):  
Ray Turner
Keyword(s):  
Recent Work ◽  
Set Theory ◽  
Semantic Theory ◽  
Truth Predicate ◽  
Minimal Logic ◽  
History Of ◽  
Concept Of Truth ◽  
Almost All ◽  
The Liar ◽  
Self Reference

Frege's attempts to formulate a theory of properties to serve as a foundation for logic, mathematics and semantics all dissolved under the weight of the logicial paradoxes. The language of Frege's theory permitted the representation of the property which holds of everything which does not hold of itself. Minimal logic, plus Frege's principle of abstraction, leads immediately to a contradiction. The subsequent history of foundational studies was dominated by attempts to formulate theories of properties and sets which would not succumb to the Russell argument. Among such are Russell's simple theory of types and the development of various iterative conceptions of set. All of these theories ban, in one way or another, the self-reference responsible for the paradoxes; in this sense they are all “typed” theories. The semantical paradoxes, involving the concept of truth, induced similar nightmares among philosophers and logicians involved in semantic theory. The early work of Tarski demonstrated that no language that contained enough formal machinery to respresent the various versions of the Liar could contain a truth-predicate satisfying all the Tarski biconditionals. However, recent work in both disciplines has led to a re-evaluation of the limitations imposed by the paradoxes.In the foundations of set theory, the work of Gilmore [1974], Feferman [1975], [1979], [1984], and Aczel [1980] has clearly demonstrated that elegant and useful type-free theories of classes are feasible. Work on the semantic paradoxes was given new life by Kripke's contribution (Kripke [1975]). This inspired the recent work of Gupta [1982] and Herzberger [1982]. These papers demonstrate that much room is available for the development of theories of truth which meet almost all of Tarski's desiderata.

A complete infinitary logic

1976 ◽  
Vol 41 (4) ◽  
pp. 730-746
Author(s):  
Kenneth Slonneger
Keyword(s):  
Natural Extension ◽  
Predicate Calculus ◽  
Cut Elimination ◽  
Infinitary Logic ◽  
Logical System ◽  
Natural Numbers ◽  
First Order ◽  
Ordinal Numbers ◽  
Power Set ◽  
Order Predicate Calculus

This paper is concerned with the proof theoretic development of certain infinite languages. These languages contain the usual infinite conjunctions and disjunctions, but in addition to homogeneous quantifiers such as ∀x0x1x2 … and ∃y0y1y2 …, we shall investigate particular subclasses of the dependent quantifiers described by Henkin [1]. The dependent quantifiers of Henkin can be thought of as partially ordered quantifiers defined by a function from one set to the power set of another set. This function assigns to each existentially bound variable, the set of universally bound variables on which it depends.The natural extension of Gentzen's first order predicate calculus to an infinite language with homogeneous quantifiers results in a system that is both valid and complete, and in which a cut elimination theorem can be proved [2]. The problem then arises of devising, if possible, a logical system dealing with general dependent quantifiers that is valid and complete. In this paper a system is presented that is valid and complete for an infinite language with homogeneous quantifiers and dependent quantifiers that are anti-well-ordered, for example, … ∀x2∃y2∀x1∃y1∀x0∃y0.Certain notational conventions will be employed in this paper. Greek letters will be used for ordinal numbers. The ordinal ω is the set of all natural numbers, and 2ω is the set of all ω -sequences of elements of 2 = {0,1}. The power set of S is denoted by P(S). μα[A(α)] stands for the smallest ordinal α such that A (α) holds.

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