scholarly journals General Recursive Functions in the Number-Theoretic Formal System

1957 ◽  
Vol 1 (2) ◽  
pp. 119-130
Author(s):  
Sh^|^ocirc;ji MAEHARA
Keyword(s):  
Formal System ◽  
Recursive Functions

Two recursion theoretic characterizations of proof speed-ups

1989 ◽  
Vol 54 (2) ◽  
pp. 522-526 ◽  
Author(s):  
James S. Royer
Keyword(s):  
Recursive Function ◽  
Formal System ◽  
Complexity Measure ◽  
Incompleteness Theorem ◽  
Similar Treatment ◽  
Recursive Functions ◽  
Partial Recursive Functions ◽  
Speed Up

Smullyan in [Smu61] identified the recursion theoretic essence of incompleteness results such as Gödel's first incompleteness theorem and Rosser's theorem. Smullyan (improving upon [Kle50] and [Kle52]) showed that, for sufficiently complex theories, the collection of provable formulae and the collection of refutable formulae are effectively inseparable—where formulae and their Gödel numbers are identified. This paper gives a similar treatment for proof speed-up. We say that a formal system S1is speedable over another system S0on a set of formulaeAiff, for each recursive functionh, there is a formulaαinAsuch that the length of the shortest proof ofαin S0is larger thanhof the shortest proof ofαin S1. (Here we equate the length of a proof with something like the number of characters making it up,notits number of lines.) We characterize speedability in terms of the inseparability by r.e. sets of the collection of formulae which are provable in S1but unprovable in S0from the collectionA–where again formulae and their Gödel numbers are identified. We provide precise definitions of proof length, speedability and r.e. inseparability below.We follow the terminology and notation of [Rog87] with borrowings from [Soa87]. Below,ϕis an acceptable numbering of the partial recursive functions [Rog87] andΦa (Blum) complexity measure associated withϕ[Blu67], [DW83].

scholarly journals Formalism and intuition in computability

2012 ◽  
Vol 370 (1971) ◽  
pp. 3277-3304 ◽  
Author(s):  
Robert Irving Soare
Keyword(s):  
Formal System ◽  
Third Wave ◽  
Turing Machines ◽  
Recursive Functions ◽  
Original Meaning ◽  
Second Wave ◽  
Informal Approach

The model of recursive functions in 1934–1936 was a deductive formal system. In 1936, Turing and in 1944, Post introduced more intuitive models of Turing machines and generational systems. When they both died prematurely in 1954, their informal approach was replaced again by the very formal Kleene T -predicate for another decade. By 1965, researchers could no longer read the papers. A second wave of intuition arose with the book by Rogers and Lachlan's revealing papers. A third wave of intuition has arisen from 1996 to the present with a return to the original meaning of computability in the sense of Turing and Gödel, and a return of ‘recursive’ to its original meaning of ‘inductive’ and the founding of Computability in Europe by Cooper and others.

N. A. Šanin. On the constructive interpretation of mathematical judgments. English translation of XXXI 255 by Elliott Mendelson. American Mathematical Society translations, ser. 2 vol. 23 (1963), pp. 109–189. - A. A. Markov. On constructive functions. English translation of XXXI 258(1) by Moshe Machover. American Mathematical Society translations, vol. 29 (1963), pp. 163–195. - S. C. Kleene. A formal system of intuitionistic analysis. The foundations of intuitionistlc mathematics especially in relation to recursive functions, by Stephen Cole Kleene and Richard Eugene Vesley, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1965, pp. 1–89. - S. C. Kleene. Various notions of realizability:The foundations of intuitionistlc mathematics especially in relation to recursive functions, by Stephen Cole Kleene and Richard Eugene Vesley, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1965, pp. 90–132. - Richard E. Vesley. The intuitionistic continuum. The foundations of intuitionistlc mathematics especially in relation to recursive functions, by Stephen Cole Kleene and Richard Eugene Vesley, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1965, pp. 133–173. - S. C. Kleene. On order in the continuum. The foundations of intuitionistlc mathematics especially in relation to recursive functions, by Stephen Cole Kleene and Richard Eugene Vesley, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1965, pp. 174–186. - S. C. Kleene. Bibliography.The foundations of intuitionistlc mathematics especially in relation to recursive functions, by Stephen Cole Kleene and Richard Eugene Vesley, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1965, pp. 187–199.

1966 ◽  
Vol 31 (2) ◽  
pp. 258-261 ◽  
Author(s):  
Georg Kreisel
Keyword(s):  
English Translation ◽  
American Mathematical Society ◽  
Formal System ◽  
Publishing Company ◽  
Mathematical Society ◽  
Foundations Of Mathematics ◽  
North Holland Publishing Company ◽  
Recursive Functions ◽  
North Holland Publishing ◽  
The Continuum

Modality and folk psychology

2019 ◽  
Vol 28 (1) ◽  
pp. 19-27
Author(s):  
Ja. O. Petik
Keyword(s):  
Modal Logic ◽  
Brain Activity ◽  
Formal System ◽  
Philosophy Of Logic ◽  
Psychological States ◽  
Formal Systems ◽  
Modern Psychology ◽  
Philosophical Interpretation ◽  
Important Direction

The connection of the modern psychology and formal systems remains an important direction of research. This paper is centered on philosophical problems surrounding relations between mental and logic. Main attention is given to philosophy of logic but certain ideas are introduced that can be incorporated into the practical philosophical logic. The definition and properties of basic modal logic and descending ones which are used in study of mental activity are in view. The defining role of philosophical interpretation of modality for the particular formal system used for research in the field of psychological states of agents is postulated. Different semantics of modal logic are studied. The hypothesis about the connection of research in cognitive psychology (semantics of brain activity) and formal systems connected to research of psychological states is stated.

Sign in / Sign up

Export Citation Format

Share Document