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DECIDABLE MODELS OF ω-STABLE THEORIES

2014
*
Journal of Symbolic Logic (JSL)
*

We further characterize the decidable models of ω-stable

doi:10.1017/jsl.2013.2
fatcat:w3463o6xjrdulj2ia7drhlyw3y
*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 . ...##
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Foundations of recursive model theory

1978
*
Annals of Mathematical Logic
*

However, Fact 3 cannot be similarly extended; we witl also show that there exists a complete decidable

doi:10.1016/0003-4843(78)90030-x
fatcat:2tcklys57nebtfxm4tz5b5vwsq
*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. ...##
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Classical descriptive set theory as a refinement of effective descriptive set theory

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

doi:10.1016/j.apal.2010.09.010
fatcat:sdgg5vwvubdr3dgwcouliocvnu
*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. ...##
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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. ...##
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From the weak to the strong existence property

2012
*
Annals of Pure and Applied Logic
*

MSC: 03F50 03F35 Keywords: Intuitionistic

doi:10.1016/j.apal.2012.01.012
fatcat:m3je6fvf4rdcviylik5ugo5lg4
*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] ). ...##
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Representing Scott sets in algebraic settings

2015
*
Archive for Mathematical Logic
*

We fix a Gödel coding of L

doi:10.1007/s00153-015-0431-1
fatcat:p4qzx5jvpfgdhpyykerwsrwmom
*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 − . ...##
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Towards Limit Computable Mathematics
[chapter]

2002
*
Lecture Notes in Computer Science
*

LCM is related not only to learning

doi:10.1007/3-540-45842-5_9
fatcat:ajugnuaa5fg5bnx7kfxq7kui2e
*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*. ...##
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Representing Scott sets in algebraic settings
[article]

2014
*
arXiv
*
pre-print

We prove that for every Scott

arXiv:1407.5612v1
fatcat:3sbukd25ljal3p7wqav6lqf4j4
*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 − . ...##
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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. ...##
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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*. ...##
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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*. ...

##
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Bibliography on Realizability

1999
*
Electronical Notes in Theoretical Computer Science
*

Abadi

doi:10.1016/s1571-0661(04)00103-3
fatcat:jpr3su4qxfgwremm5yjfgwbgai
*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. ...##
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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. ...##
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Goodman's theorem and beyond

1979
*
Pacific Journal of Mathematics
*

We can therefore assume that T is one of the nonextensional

doi:10.2140/pjm.1979.84.1
fatcat:dljomcrlcbegvo3vk4v4gatszq
*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 ...##
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Type Structure Complexity and Decidability

1982
*
Transactions of the American Mathematical Society
*

Specifically, the principal result of this paper is to prove that if â is homogeneous

doi:10.2307/1998751
fatcat:pknfy6567zcaho4uszrzwngr5q
*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. ...
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