Undecidable Theories Research Articles

Π⁰₁ classes and degrees of theories

Using the methods of recursive function theory we derive several results about the degrees of unsolvability of members of certain Π 1 0 \Pi _1^0 classes of functions (i.e. degrees of branches of certain recursive trees). As a special case we obtain information on the degrees of consistent extensions of axiomatizable theories, in particular effectively inseparable theories such as Peano arithmetic, P {\mathbf {P}} . For example: THEOREM 1. If a degree a {\mathbf {a}} contains a complete extension of P {\mathbf {P}} , then every countable partially ordered set can be embedded in the ordering of degrees ⩽ a \leqslant {\mathbf {a}} . (This strengthens a result of Scott and Tennenbaum that no such degree a {\mathbf {a}} is a minimal degree.) THEOREM 2. If T {\mathbf {T}} is an axiomatizable, essentially undecidable theory, and if { a n } \{ {{\mathbf {a}}_n}\} is a countable sequence of nonzero degrees, then T {\mathbf {T}} has continuum many complete extensions whose degrees are pairwise incomparable and incomparable with each a n {{\mathbf {a}}_n} . THEOREM 3. There is a complete extension T {\mathbf {T}} of P {\mathbf {P}} such that no nonrecursive arithmetical set is definable in T {\mathbf {T}} . THEOREM 4. There is an axiomatizable, essentially undecidable theory T {\mathbf {T}} such that any two distinct complete extensions of T {\mathbf {T}} are Turing incomparable. THEOREM 5. The set of degrees of consistent extensions of P {\mathbf {P}} is meager and has measure zero.

The use of lists in the study of undecidable problemsin automata theory

Many of the problems in automata theory are unsolvable and can be classified intodegrees of unsolvability by their relative difficulty. In this note, natural reference sets are presented which belong to the complete degrees at each level of the arithmetic hierarchy. Also, some questions regarding lists of recursively enumerable sets are considered. These results resolve some apparent peculiarities and provide simple methods of determining the degrees of unsolvability for several well-known problems and permit easy construction of natural problems with high degrees of unsolvability.

Axiomatizable theories with few axiomatizable extensions

In this paper we prove two theorems. They answer questions raised by Myhill in 1956. (We recall the well-known fact that Myhill's invention of the maximal set in 1956 [2] stemmed from his attempt to prove I below.)I. There exists an axiomatizable, essentially undecidable theory in standard formalization such that all axiomatizable extensions of are finite extensions.II. There exists an axiomatizable but undecidable theory in standard formalization such that(a) has a consistent, complete, decidable extension ,(b) If is an axiomatizable extension of then either(i) is a finite extension of , or(ii) is a finite extension of .

Robert L. Vaught. On a theorem of Cobham concerning undecidable theories. Logic, methodology and philosophy of science, Proceedings of the 1960 International Congress, edited by Ernest Nagel, Patrick Suppes, and Alfred Tarski, Stanford University Press, Stanford, Calif., 1962, pp. 14–25.

Robert L. Vaught. On a theorem of Cobham concerning undecidable theories. Logic, methodology and philosophy of science, Proceedings of the 1960 International Congress, edited by Ernest Nagel, Patrick Suppes, and Alfred Tarski, Stanford University Press, Stanford, Calif., 1962, pp. 14–25. - Volume 34 Issue 1

Solution to a problem of Rose and Rosser

2. R. Montague, Semantical closure and non-finite axiomatizability. I, Proceedings of the 1959 International Symposium on the Foundations of Mathematics: Infinitistic Methods, to appear. 3. K. Godel, The consistency of the continuum hypothesis, Princeton, N. J., Princeton University Press, 1953. 4. I. L. Novak, A constructson for models of consistent systems, Fund. Math. vol. 37 (1950) pp. 87-110. 5. J. R. Shoenfield, A relative consistency proof, J. Symb. Logic vol. 19 (1954) pp. 21-28. 6. A. Tarski, A. Mostowski and R. M. Robinson, Undecidable theories, Amsterdam, North-Holland Publishing Co., 1953.

Reviews - Alfred Tarski. Preface. Undecidable theories, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1953, pp. VIII–IX. - Alfred Tarski. A general method in proofs of undecidability. Undecidable theories, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1953, pp. 3–35. - Andrzej Mostowski, Raphael M. Robinson, and Alfred Tarski. Undecidability and essential undecidability

Reviews - Alfred Tarski. Preface. Undecidable theories, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1953, pp. VIII–IX. - Alfred Tarski. A general method in proofs of undecidability. Undecidable theories, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1953, pp. 3–35. - Andrzej Mostowski, Raphael M. Robinson, and Alfred Tarski. Undecidability and essential undecidability in arithmetic. Undecidable theories, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1953, pp. 39–74. - Alfred Tarski. Undecidability of the elementary theory of groups. Undecidable theories, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1953, pp. 77–87. - Bibliography. Undecidable theories, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1953, pp. 89–91. - Index. Undecidable theories, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1953, pp. 93–98. - Volume 24 Issue 2

Undecidable Theories. By Alfred Tarski in collaboration with Andrzej Mostowski and Raphael M. Robinson. North-Holland Publishing Co., Amsterdam, 1953. Pp. xxi + 98. 18s.

Previous articleNext article No AccessREVIEWSUndecidable Theories. By Alfred Tarski in collaboration with Andrzej Mostowski and Raphael M. Robinson. North-Holland Publishing Co., Amsterdam, 1953. Pp. xxi + 98. 18s.PAUL BERNAYSPAUL BERNAYS Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by The British Journal for the Philosophy of Science Volume 9, Number 36February 1959 Society: The British Society for the Philosophy of Science Views: 5Total views on this site Views: 5Total views on this site Article DOIhttps://doi.org/10.1093/bjps/IX.36.321 Views: 5Total views on this site © 1959 by The Author. All rights reserved.PDF download Crossref reports no articles citing this article.

Two theories with axioms built by means of pleonasms

This paper contains examples T1 and T2 of theories which answer the following questions:(1) Does there exist an essentially undecidable theory with a finite number of non-logical constants which contains a decidable, finitely axiomatizable subtheory?(2) Does there exist an undecidable theory categorical in an infinite power which has a recursive set of axioms? (Cf. [2] and [3].)The theory T1 represents a modification of a theory described by Myhill [7]. The common feature of theories T1 and T2 is that in both of them pleonasms are essential in the construction of the axioms.Let T1 be a theory with identity = which contains one binary predicate R(x, y) and is based on the axioms A1, A2, A3, B1, B2, B3, B4, Cnm which follow.A1: x = x. A2: x = y ⊃ y = x. A3: x = y ∧ y = z ⊃ x = z.(Axioms of identity.)B1: R(x, x). B2: R(x, y) ⊃ R(y, x). B3: R (x, y) ∧ R(y, z) ⊃ R(x, z).(Axioms of equivalence.)B4: x = y ⊃ [R(z, x) ≡ R(z,y)].Let φn be the formulawhich express that there is an abstraction class of the relation R which has exactly n elements.Let f(n) and g(n) be two recursive functions which enumerate two recursively inseparable sets [5], and call these sets X1 and X2.We now specify the axioms Cmm.It is obvious that the set composed of the formulas A1−A3, B1−B4, Cnm (n,m = 1,2, …) is recursive.The theory T1 is essentially undecidable; for if there were a complete and decidable extension T′1 (of it, then the recursive sets Z = {n: φn is provable in T′1} and Z′ = {n: ∼φn is provable in T′1} would separate the sets X1 and X2.

Decidability and essential undecidability

There are a number of open problems involving the concepts of decidability and essential undecidability. This paper will present solutions to some of these problems. Specifically:(1) Can a decidable theory have an essentially undecidable, axiomatizable extension (with the same constants)?(2) Are all the complete extensions of an undecidable theory ever decidable?We shall show that the answer to both questions is in the affirmative. In answering question (1), the decidable theory for which an essentially undecidable axiomatizable extension will be constructed is the theory of the successor function and a single one-place predicate. It will also be shown that the decidability of this theory is a “best possible” result in the following direction: the theory of either of the common diadic arithmetic functions and a one-place predicate; i.e., of addition and a one-place predicate, or of multiplication and a one-place predicate, is undecidable.Before establishing the main result, it is convenient to give a simple proof that a decidable theory can have an axiomatizable (simply) undecidable extension. This is, of course, an immediate consequence of the main result; but the proof is simple and illustrates the methods that we are going to use in this paper.

Solution of a problem of Tarski

We presuppose the terminology of [1], and we give a negative answer to the following problem ([1], p. 19): Does every essentially undecidable axiomatizable theory have an essentially undecidable finitely axiomatizable subtheory?We use the following theorem of Kleene ([2], p. 311). There exist two recursively enumerable sets α and β such that (1) α and β are disjoint (2) there is no recursive set η for which α ⊂ η, β ⊂ η′. By the definition of recursive enumerability, there are recursive predicates Φ and Ψ for whichWe now specify a theory T which will afford a counter-example to the given problem of Tarski. The only non-logical constants of T are two binary predicates P and Q, one unary operation symbol S, and one individual constant 0. As in ([1], p. 52) we defineThe only non-logical axioms of T are the formulae P(Δm, Δn) for all pairs of integers m, n satisfying Δ(m, n); the formulae Q(Δm, Δn) for all pairs of integers m, n satisfying Ψ(m, n); and the formulaT is consistent, since it has a model. It remains to show that (1) every consistent extension of T is undecidable (2) if T1 is a finitely axiomatizable subtheory of T, there exists a consistent and decidable extension of T1 which has the same constants as T1.

Undecidable Theories. By Alfred Tarski in collaboration with Andrzej Mostowski and Raphael M. Robinson. (Amsterdam: North-Holland Publishing Company. 1953. Pp. 98. Price 18s.)

Undecidable Theories. By Alfred Tarski in collaboration with Andrzej Mostowski and Raphael M. Robinson. (Amsterdam: North-Holland Publishing Company. 1953. Pp. 98. Price 18s.)

Book Review: Undecidable theories

Book Review: Undecidable theories

A remark concerning decidability of complete theories

A formalized theory is called complete if for each sentence expressible in this theory either the sentence itself or its negation is provable.A theory is called deciddble if there exists an effective procedure (called decision-procedure) which enables one to decide of each sentence, in a finite number of steps, whether or not it is provable in the theory.It is known that there exist complete but undecidable theories. There exist, namely, the so called essentially undecidable theories, i.e. theories which are undecidable and remain so after an arbitrary consistent extension of the set of axioms. Using the well-known method of Lindenbaum we can therefore obtain from each such theory a complete and undecidable theory.The aim of this paper is to prove a theorem which shows that complete theories satisfying certain very general conditions are always decidable. In somewhat loose formulation these conditions are: There exist four effective methods M1, M2, M3, M4, such that(a) M1 enables us to decide in each case whether or not any given formula is a sentence of the theory;(b) M2 gives an enumeration of all axioms of the theory;(c) the rules of inference can be arranged in a sequence R1, R2, … such that if p1, … pk, r are arbitrary sentences of the theory, we can decide by M3 whether or not r results from p1, … pk, by the n-th rule;(d) M4 enables us to construct effectively the negation of each effectively given sentence.In order to express these conditions more precisely we shall make use of an arithmetization of the considered theory .