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2014 Journal of Symbolic Logic (JSL)  
We further characterize the decidable models of ω-stable theories with countably many countable models as those which realize only recursive types.  ...  In particular, we show that for a countable ω-stable T, every countable model of T admits a decidable presentation if and only if all n-types in T are recursive and T has only countably many countable  ...  Let T be a complete decidable theory, and let Ψ be a recursively enumerable set of non-principal (not necessarily complete) types in T .  ... 
doi:10.1017/jsl.2013.2 fatcat:w3463o6xjrdulj2ia7drhlyw3y

Foundations of recursive model theory

Terrence S. Millar
1978 Annals of Mathematical Logic  
However, Fact 3 cannot be similarly extended; we witl also show that there exists a complete decidable theory T and recursive types F~, F2 of T such that no decidable model of T realizes both F~ and r~  ...  Then 92 is decidable just in case, for some indexing E of 192[, {n 192~0~ is a recursive set of integers.  ...  Let T be a c,"mplete decidable theory and 9.I an homogeneous model of T realizing only recursive types of T.  ... 
doi:10.1016/0003-4843(78)90030-x fatcat:2tcklys57nebtfxm4tz5b5vwsq

Classical descriptive set theory as a refinement of effective descriptive set theory

Yiannis N. Moschovakis
2010 Annals of Pure and Applied Logic  
The (effective) Suslin-Kleene Theorem is obtained as a corollary of a standard proof of the classical Suslin Theorem, by noticing that it is mostly constructive and applying to it a naive realizability  ...  I thank Peter Aczel, Jeremy Avigad, Thierry Coquand, Ulrich Kohlenbach, Andre Scedrov, Jaap van Oosten, Michael Rathjen and Wim Veldman, for useful comments on an early draft of this article, and I am  ...  especially grateful to my wife Joan Rand Moschovakis for listening patiently and responding usefully to many questions (some of them not very coherent) on a subject that she knows well and I do not.  ... 
doi:10.1016/j.apal.2010.09.010 fatcat:sdgg5vwvubdr3dgwcouliocvnu

Page 22 of Mathematical Reviews Vol. , Issue 81A [page]

1981 Mathematical Reviews  
R, denotes the set of all realizable formulas. Necessary and suffi- cient conditions for the expressibility in R, of predicates defined on recursive sets are given.  ...  Berezin (Novosibirsk) Prank, R. 81a:03048 Expressibility in the elementary theory of recursive sets with realizability logic. (Russian. English summary) Tartu Riikl. Ul.  ... 

From the weak to the strong existence property

Michael Rathjen
2012 Annals of Pure and Applied Logic  
MSC: 03F50 03F35 Keywords: Intuitionistic set theory Collection axiom Realizability with sets of witnesses Weak existence property Set recursive functions a b s t r a c t A hallmark of many an intuitionistic  ...  semantics for constructive Zermelo-Fraenkel set theory, CZF, that combines realizability for extensional set theory and truth.  ...  An even more important reason is that admissible sets have been a major source of interaction between model theory, recursion theory and set theory (cf. [4] ).  ... 
doi:10.1016/j.apal.2012.01.012 fatcat:m3je6fvf4rdcviylik5ugo5lg4

Representing Scott sets in algebraic settings

Alf Dolich, Julia F. Knight, Karen Lange, David Marker
2015 Archive for Mathematical Logic  
We fix a Gödel coding of L and say that a set of L-formulas is in S if the corresponding set of Gödel codes is in S. Let T be a complete L-theory.  ...  This can be done recursively in tp(b, a) and tp(a, c). Thus, every type realized in H is in S. Case 2: Suppose b ∈ G realizes p − .  ... 
doi:10.1007/s00153-015-0431-1 fatcat:p4qzx5jvpfgdhpyykerwsrwmom

Towards Limit Computable Mathematics [chapter]

Susumu Hayashi, Masahiro Nakata
2002 Lecture Notes in Computer Science  
LCM is related not only to learning theory and recursion theory, but also to many areas in mathematics and computer science such as computational algebra, computability theories in analysis, reverse mathematics  ...  The computational content of LCM-proofs is given by Gold's limiting recursive functions, which is the fundamental notion of learning theory.  ...  A realizability and formal system A formal semantics of LCM is rst given according to Kleene realizability interpretation by \taking the limit" of basic recursive function theory.  ... 
doi:10.1007/3-540-45842-5_9 fatcat:ajugnuaa5fg5bnx7kfxq7kui2e

Representing Scott sets in algebraic settings [article]

Alf Dolich, Julia Knight, Karen Lange, David Marker
2014 arXiv   pre-print
We prove that for every Scott set S there are S-saturated real closed fields and models of Presburger arithmetic.  ...  We fix a Gödel coding of L and say that a set of L-formulas is in S if the corresponding set of Gödel codes is in S. Let T be a complete L-theory.  ...  This can be done recursively in tp(b, a) and tp(a, c). Thus, every type realized in H is in S. Case 2: Suppose b ∈ G realizes p − .  ... 
arXiv:1407.5612v1 fatcat:3sbukd25ljal3p7wqav6lqf4j4

Page 2781 of Mathematical Reviews Vol. , Issue 88f [page]

1988 Mathematical Reviews  
Unlike the analogy between recursive set theory and classical set theory, the analogy between recursive set theory and construc- tive set theory is exceptionless.  ...  Leonard Lipshitz (1-PURD) 88f:03038 03D50 03E70 03F55 McCarty, Charles (4-EDIN-C) Realizability and recursive set theory. Ann. Pure Appl. Logic 32 (1986), no. 2, 153-183.  ... 

Page 3176 of Mathematical Reviews Vol. , Issue 2003e [page]

2003 Mathematical Reviews  
They extend our previous finite axiomatizations of inductive-recursive definitions of sets to indexed families of sets and encompass vir- tually all definitions of sets which have been used in intuitionistic  ...  Summary: “We give two finite axiomatizations of indexed inductive-recursive definitions in intuitionistic type theory.  ... 

Page 1 of Mathematical Reviews Vol. 27, Issue 5 [page]

1964 Mathematical Reviews  
set theory, then the consistency of 7' is demonstrable in Zermelo-Fraenkel set theory, and hence 7' does not coincide with the Zermelo-Fraenkel set theory.  ...  The reviewer and the authors have subsequently observed (independently) that this attempt fails. H. Putnam (Cambridge, Mass.) Montague, Richard Two contributions to the foundations of set theory.  ... 

Bibliography on Realizability

Lars Birkedal
1999 Electronical Notes in Theoretical Computer Science  
Abadi and L. Cardelli. A Theory of Objects. Springer Verlag, 1996. 2] M. Abadi and G.D. Plotkin. A per model of polymorphism and recursive t ypes. In  ...  The consistency of some intuitionistic and constructive principles with a set theory. Studia Logica, 40:237{248, 1981. 139] V. Kh. Khakhanyan. Set theory and Church's thesis (Russian). In A. I.  ...  The computability of primitive recursive terms of nite type, and primitive recursive realization (Russian). Zapiski, 8:32{45, 1968. Translation Seminars in Mathematics.  ... 
doi:10.1016/s1571-0661(04)00103-3 fatcat:jpr3su4qxfgwremm5yjfgwbgai

Page 1207 of Mathematical Reviews Vol. 35, Issue 6 [page]

1968 Mathematical Reviews  
The partial recursive function ¢ realizes P if, for every k, y(k) is defined and realizes P(A,', ---, A,").  ...  Quine, Set theory and its logic, Harvard Univ. Press, Cambridge, Mass., 1963; K. R. Brown and the author, J. Symbolic Logic 31 (1966), 409-414; MR 34 #1163; and W. Quine and the author, Bull. Amer.  ... 

Goodman's theorem and beyond

Michael Beeson
1979 Pacific Journal of Mathematics  
We can therefore assume that T is one of the nonextensional set theories discussed in [3] . Then forcing and realizability are both sound for T, as proved in [3] .  ...  Also in the case of Friedman's set theories, there need to be certain constants to denote the sets whose existence is asserted by the axioms, and certain function symbols, and we have to give c* for terms  ... 
doi:10.2140/pjm.1979.84.1 fatcat:dljomcrlcbegvo3vk4v4gatszq

Type Structure Complexity and Decidability

T. S. Millar
1982 Transactions of the American Mathematical Society  
Specifically, the principal result of this paper is to prove that if â is homogeneous and the set of recursive types of the theory of 6E is 22, then & is decidable iff the set of types realized by 6B is  ...  If 6B is homogeneous and realizes either no nonprincipal types [1] or all recursive types [2] then , . & is decidable iff the set of types realized by 6B is an r.e. (*) set of recursive types.  ...  Every type of a complete theory T which is recursive in the degree of T is realized in every recursively saturated model of T.  ... 
doi:10.2307/1998751 fatcat:pknfy6567zcaho4uszrzwngr5q
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