Skip to main content Accessibility help
Admissible Sets and Structures
  • Cited by 3
  • Export citation
  • Recommend to librarian
  • Buy the print book

Book description

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. Admissible set theory is a major source of interaction between model theory, recursion theory and set theory, and plays an important role in definability theory. In this volume, the seventh publication in the Perspectives in Logic series, Jon Barwise presents the basic facts about admissible sets and admissible ordinals in a way that makes them accessible to logic students and specialists alike. It fills the artificial gap between model theory and recursion theory and covers everything the logician should know about admissible sets.

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.


Aczel, P. 1970 Implicit and inductive definability (abstract). J. Symbolic Logic 35, 599 (1970).
Aczel, P., Richter, W. 1973 Inductive definitions and reflecting properties of admissible ordinals. In: Fenstad, J. E. and Hinman, P., eds., Generalized Recursion Theory, pp. 301—381. Amsterdam: North-Holland 1973.
Barwise, J. 1967 Infinitary Logic and Admissible Sets. Ph. D. Thesis. Stanford, CA: Stanford Univ. 1967.
Barwise, J. 1968 The Syntax and Semantics of Infinitary Logic, editor (Lecture Notes in Math., Vol. 72). Berlin-Heidelberg-New York: Springer 1968.
Barwise, J. 1969a Infinitary Logic and Admissible Sets. J. Symbolic Logic 34, 226—252 (1969).
Barwise, J. 1969b Applications of Strict π1 1 predicates to Infinitary Logic, J. Symbolic Logic 34, 409—423 (1969).
Barwise, J. 1971 Infinitary Methods in the Model Theory of Set Theory, Logic Colloquium 69, pp. 53—66. Amsterdam: North-Holland 1971.
Barwise, J. 1973a Back and Forth Through Infinitary Logic, Studies in Model Theory, pp. 5—34, edited by M. Morley. MAA Studies in Mathematics 1973.
Barwise, J. 1973b Abstract Logics and L∞ω , Ann. of Math. Logic 4, 309—340 (1973).
Barwise, J. 1973c A Preservation Theorem for Interpretations, Proceedings of the Cambridge Logic Conference, pp. 618—621 (Lecture Notes in Math., Vol. 337). Berlin-Heidelberg-New York: Springer 1973.
Barwise, J. 1974a Admissible Sets over Models of Set Theory, Generalized Recursion Theory, pp. 97—122. Amsterdam: North-Holland 1974.
Barwise, J. 1974b Mostowski's Collapsing Function and the Closed Unbounded Filter. Fund. Math. 82, 95—103 (1974).
Barwise, J., Fisher, E. 1970 The Shoenfield Absoluteness Lemma. Israel J. Math. 8, 329—339 (1970).
Barwise, J., Gandy, R., Moschovakis, Y. 1971 The next admissible set. J. Symbolic Logic 36, 108—120 (1971).
Barwise, J., Kunen, K. 1971 Hanf numbers for fragments of L∞ω ,. Israel J. Math. 10, 306—320 (1971).
Chang, C. C. 1964 Some new results in definability. Bull. Amer. Math. Soc. 70, 808—813 (1964).
Chang, C. C. 1968 Some remarks on the model theory of Infinitary languages. In: Barwise [1968], pp. 36—63.
Chang, C. C, Moschovakis, Y. N. 1968 On Σ1 1-relations on special models. Notices Amer. Math. Soc. 15, 934 (1968).
Chang, C. C, Moschovakis, Y. N. 1970 The Suslin-Kleene theorem for Vκ with cofinality (κ) = ω. Pacific J. Math. 35, 565—569 (1970).
Church, A. 1938 The constructive second number class. Bull. Amer. Math. Soc. 44, 224—232 (1938).
Church, A., Kleene, S. C. 1937 Formal definitions in the theory of ordinal numbers. Fund. Math. 28, 11—21 (1937).
Cutland, N. 1972 Σ1-compactness and ultraproducts. J. Symbolic Logic 37, 668—672 (1972).
Devlin, K. V. 1973 Aspects of constructibility (Lecture Notes in Math., Vol. 354). Berlin-Heidelberg-New York: Springer 1973.
Dickmann, M. A. 1970 Model theory of infinitary languages (Aarhus University Lecture Notes, series No. 20) 1970.
Ehrenfeucht, A., Kreisel, G. 1966 Strong Models of Arithmetic. Bull. Acad. Polon. Sci. 14, 107—110 (1966).
Ehrenfeucht, A., Mostowski, A. 1956 Models admitting automorphisms. Fund. Math. 43, 50—63 (1956).
Enderton, H. 1972 A Mathematical Introduction to Logic. New York and London: Academic Press 1972.
Engeler, E. 1961 Unendliche Formeln in der Modell-Theorie. Z. Math. Logik Grundlagen Math. 7, 154—160 (1961).
Feferman, S. 1965 Some applications of forcing and generic sets. Fund. Math. 56, 325—345 (1965).
Feferman, S. 1968 Lectures on Proof Theory, Proceedings of the Summer School in Logic, Leeds 1967 (Lecture Notes in Math., Vol. 70). Berlin-Heidelberg-New York: Springer 1968.
Feferman, S. 1974 Some predicative set theories, Axiomatic Set Theory II. Amer. Math. Soc, Providence, R.I. (1974).
Feferman, S., Kreisel, G. 1966 Persistent and invariant formulas relative to theories of higher order. Bull. Amer. Math. Soc. 72, 480—485 (1966).
Flum, J. 1972 Hanf numbers and well-ordering numbers. Arch. Math. Logik Grundlagenforsch. 15,164—178 (1972).
Friedman, H. 1973 Countable models of set theories, Cambridge Summer School in Math. Logic, pp. 539—573 (Lecture Notes in Math., Vol. 337). Berlin-Heidelberg-New York: Springer 1973.
Friedman, H., Jensen, R. 1968 Note on admissible ordinals. In: Barwise [1968], pp. 77—79.
Gaifman, H. 1970 On local arithmetical functions and their application for constructing models of Peano's arithmetic, Mathematical Logic and Foundations of Set Theory (Y. Bar-Hillel, ed.), pp. 105—121. Amsterdam: North-Holland 1970.
Gale, D., Stewart, F. M. 1953 Infinite games of perfect information. Ann. Math. Studies 28, 245—266 (1953).
Gandy, R. O. 1974 Inductive definitions, Generalized Recursion Theory, pp. 265—300. Amsterdam: North- Holland 1974.
Gandy, R. O. 1975 Basic functions, Axiomatic Set Theory II. Amer. Math. Soc, Providence, R. I. (1974).
Gandy, R., Kriesel, G., Tait, W. 1960 Set existence. Bull. Acad. Polon. Ser. Sci. Math. Astron. Phys. 8, 577—582 (1960).
Garland, S. J. 1972 Generalized interpolation theorems. J. Symbolic Logic 37, 343—351 (1972).
Gödel, K. 1930 Die Vollständigkeit der Axiome des logischen Funktionenkalküls. Monatsh. Math. Phys. 37, 349—360 (1930).
Gödel, K. 1931 Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatsh. Math. Phys. 38, 173—198 (1931).
Gödel, K. 1939 Consistency proof for the generalized continuum hypothesis. Proc. Natl. Acad. Sci. U.S.A. 25, 220—224 (1939).
Gödel, K. 1940 The consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory. Annals Math. Studies 3 (Princeton, N. J., Princeton Univ. Press).
Gordon, C. 1970 Comparisons between some generalizations of recursion theory. Compositio Math. 22, 333—346 (1970).
Gordon, C. 1971 Finitistically computable functions and relations on an abstract structure. J. Symbolic Logic 36, 704 (1971).
Grilliot, T. 1971 Inductive definitions and computability. Trans. Amer. Math. Soc. 158, 309—317 (1971).
Grilliot, T. 1972 Omitting types; applications to recursion theory. J. Symbolic Logic 37, 81—89 (1972).
Grzegorczyk, A., Mostowski, A., Ryll-Nardzewski, C. 1958 The classical and ω-complete arithmetic. J. Symbolic Logic 23, 188—206 (1958).
Grzegorczyk, A., Mostowski, A., Ryll-Nardzewski, C. 1961 Definability of sets in models of axiomatic theories. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 9, 163—167 (1961).
Hanf, W. 1964 Incompactness in languages with infinitely long expressions. Fund. Math. LIII, 309—324 (1964).
Hanf, W., Scott, D. 1961 Classifying inaccessible cardinals. Notices Amer. Math. Soc. 8, 445 (1961).
Harrison, J. 1966 Recursive pseudo-wellorderings. Ph. D.Thesis. Stanford, CA: Stanford Univ.
Henkin, L. 1949 The completeness of the first-order predicate calculus. J. Symbolic Logic 14, 159—166 (1949).
Henkin, L. 1954 A generalization of the concept of co-consistency. J. Symbolic Logic 19, 183—196 (1954).
Henkin, L. 1957 A generalization of the concept of co-completeness. J. Symbolic Logic 22, 1—14 (1957).
Jech, T. 1973 Some combinatorial problems concerning uncountable cardinals. Annals of Math. Logic 5, 165—198 (1973).
Jensen, R. B. 1972 The fine structure of the constructible hierarchy. Annals of Math. Logic 4, 229-308 (1972).
Jensen, R., Karp, C. 1971 Primitive recursive set functions, Axiomatic Set Theory. Proceedings of the set theory institute, UCLA, 1967, Amer. Math. Soc. (1971), Providence, R.I.
Karp, C. 1962 Independence proofs in predicate logic with infinitely long expressions. J. Symbolic Logic 27, 171—188 (1962).
Karp, C. 1964 Languages with Expressions of Infinite Length. Amsterdam: North-Holland 1964.
Karp, C. 1965 Finite quantifier equivalence, The Theory of Models. (Edited by J. Addison, L. Henkin and A. Tarski) pp. 407—412. Amsterdam: North-Holland 1965.
Karp, C. 1967 Nonaxiomatizability results for infinitary systems. J. Symbolic Logic 32, 367—384 (1967).
Karp, C. 1968 An algebraic proof of the Barwise compactness theorem. In: Barwise [1968] pp. 80—95.
Karp, C. 1972 From countable to cofinality co in infinitary model theory. J. Symbolic Logic 37, 430 (1972).
Keisler, H. J. 1965 Finite approximations of infinitely long formulas, The Theory of Models (edited by J. Addison, L. Henkin and A. Tarski) pp. 158—169. Amsterdam: North-Holland 1965.
Keisler, H. J. 1968 Formulas with linearly ordered quantifiers. In: Barwise [1968], pp. 96—130.
Keisler, H. J. 1971 Model Theory for Infinitary Logic. Amsterdam: North-Holland.
Keisler, H. J. 1973 Forcing and the omitting types theorem, Studies in Model Theory, MAA (1973).
Kino, A., Takeuti, G. 1962 On Hierarchies of predicates of ordinal numbers. J. Math. Soc. Japan 14, 199—232 (1962).
Kleene, S. C. 1938 On notations for ordinal numbers. J. Symbolic Logic 3, 150—155 (1938).
Kleene, S. C. 1944 On the forms of the predicates in the theory of constructive ordinals. Amer. J. Math. 66, 41—58 (1944).
Kleene, S. C. 1955a On the forms of the predicates in the theory of constructive ordinals [second paper]. Amer. J. Math. 77, 405—428 (1955).
Kleene, S. C. 1955b Arithmetical predicates and function quantifiers. Trans. Amer. Math. Soc. 79, 312—340 (1955).
Kleene, S. C. 1955c Hierarchies of number theoretic predicates. Bull. Amer. Math. Soc. 61, 193—213 (1955).
Kleene, S. C. 1959a Quantification of number theoretic functions. Compositio Math. 14, 23—40 (1959).
Kleene, S. C. 1959b Recursive functional and quantifiers of finite types I. Trans. Amer. Math. Soc. 91, 1—52 (1959).
Kreisel, G. 1961 Set theoretic problems suggested by the notion of potential totality. In: Infinitistic Methods, pp. 103—140. Oxford: Pergamon 1961.
Kreisel, G. 1962 The axiom of choice and the class of hyperarithmetic functions. Indag. Math. 24, 307—319 (1962).
Kreisel, G. 1965 Model theoretic invariants: applications to recursive and hyperarithmetic operatons. In: Addison, J. et al. (eds.) Theory of Models (Proceedings of the 1963 Berkeley Symposium) pp. 190—205. Amsterdam: North-Holland 1965.
Kreisel, G. 1968 Choice of infinitary language by means of definability criteria. In: Barwise [1968] pp. 139—151.
Kreisel, G. 1971 Some reasons for generalizing recursion theory. In: Gandy, R. O. and Yates, C. E. M. (eds.) Logic Colloquium 69, pp.139—198. Amsterdam: North-Holland 1971.
Kreisel, G., Krivine, J. L. 1967 Elements of mathematical logic. Amsterdam: North-Holland 1967.
Kreisel, G., Sacks, G. E. 1965 Metarecursive sets. J. Symbolic Logic 30, 318—338 (1965).
Kripke, S. 1964 Transfinite recursion on admissible ordinals, I, II (abstracts). J. Symbolic Logic 29, 161—162 (1964).
Krivine, J. L., McAloon, K. 1973 Some true unprovable formulas for set theory. Proceedings of the Bertrand Russell Logic Conference, Denmark 1971, pp. 332—341. Leeds 1973.
Kueker, D. 1968 Definability, automorphisms, and infinitary languages. In: Barwise [1968] pp. 152—165.
Kueker, D. 1970 Generalized interpolation and definability. Annals Math. Logic 1, 423—468 (1970).
Kueker, D. 1972 Lowenheim-Skolem and interpolation theorems in infinitary languages. Bull. Amer. Math. Soc. 78, 211—215(1972).
Kunen, K. 1968 Implicit definability and infinitary languages. J. Symbolic Logic 33, 446—451 (1968).
Lévy, A. 1963 Transfinite computability, Notices Amer. Math. Soc. 10, 286 (1963).
Lévy, A. 1965 A hierarchy of formulas in set theory. Memoir Amer. Math. Soc. 57 (1965).
Lopez-Escobar, E. 1965 An interpolation theorem for denumerably long sentences. Fund. Math. LVII, 253—272 (1965).
Lopez-Escobar, E. 1966 On definable well-orderings. Fund. Math. LVIX, 13—21 and 299—300 (1966).
Machover, M. 1961 The theory of transfinite recursion. Bull. Amer. Math. Soc. 67, 575—578 (1961).
Makkai, M. 1964 On a generalization of a theorem of E. W. Beth. Acta Math. Acad. Sci. Hungarica 15, 227—235 (1964).
Makkai, M. 1969 An application of a method of Smullyan to logics on admissible sets. Bull, de l'Académie Polon. des Sci., Ser. Math. 17, 341—346 (1969).
Makkai, M. 1973 Global defmibility theory in Lω1ω. Bull. Amer. Math. Soc. 79, 916—921 (1973).
Makkai, M. 1975 Applications of a result on weak definability theory in Lω1ω (to appear).
Malitz, J. 1965 Problems in the model theory of infinite languages, Ph. D. Thesis. Berkeley: Univ. of California 1965.
Malitz, J. 1971 Infinitary analogues of theorems from first order model theory. J. Symbolic Logic 36, 216—228 (1971).
Montague, R. 1968 Recursion theory as a branch of model theory. In: van Rootselaar, B. et al. (eds.) Logic, Methodology and Philosophy of Science III (Proceedings of the 1967 Congress) pp. 63—86. Amsterdam: North-Holland 1968.
Morley, M. 1965 Omitting classes of elements, The Theory of Models (J. W. Addison, L. Henkin and A. Tarski, eds.) pp. 265—273. Amsterdam: North-Holland 1965.
Morley, M. 1967 The Hanf number for co-logic (abstract). J. Symbolic Logic 32, 437 (1967).
Morley, M. 1970 The number of countable models. J. Symbolic Logic 35, 14—18 (1970).
Morley, M., Morley, V. 1967 The Hanf number for K-logic (abstract). Notices Amer. Math. Soc. 14, 556 (1967).
Moschovakis, Y. N. 1969a Abstract first order computability I. Trans. Amer. Math. Soc. 138, 427—464 (1969).
Moschovakis, Y. N. 1969b Abstract first order computability II. Trans. Amer. Math. Soc. 138, 465—504 (1969).
Moschovakis, Y. N. 1970 The Suslin-Kleene theorem for countable structures. Duke Math. J. 37, 341—352 (1970).
Moschovakis, Y. N. 1971 The game quantifier, Proc. Amer. Math. Soc. 31, 245—250 (1971).
Moschovakis, Y. N. 1974 Elementary Induction on Abstract Structures. Amsterdam: North-Holland 1974.
Mostowski, A. 1949 An undecidable arithmetical statement. Fund. Math. 36, 143—164 (1949).
Mostowski, A. 1961 Formal system of analysis based on an infinitistic rule of proof, Infinitistic Methods. Warsaw 1961.
Nadel, M. 1971 Model Theory in Admissible Sets, Ph. D. Thesis. Univ. of Wisconsin 1971.
Nadel, M. 1972a Some Löwenheim-Skolem results for admissible sets. Israel J. Math. 12, 427 (1972).
Nadel, M. 1972b An application of set theory to model theory. Israel J. Math. 11, 386 (1972).
Nadel, M. 1974 Scott sentences and admissible sets. Annals of Math. Logic 7, 267—294 (1974).
Nerode, A. 1957 General topology and partial recursive functions, Summaries of talks presented at the Summer Institute for Symbolic Logic, pp. 247—251. Cornell Univ. 1957.
Orey, S. 1956 On ω-consistency and related properties. J. Symbolic Logic 21, 246—252 (1956).
Platek, R. 1966 Foundations of recursion theory, Doctoral Dissertation and Supplement. Stanford, CA: Stanford Univ. 1966.
Reyes, G. E. 1970 Local definability theory. Annals of Math. Logic 1, 95—137 (1970).
Richter, W. 1971 Recursively Mahlo ordinals and inductive definitions. In: Gandy, R. O. and Yates, C. E. M. (eds.) Logic Colloquium '69, pp. 273—288. Amsterdam: North-Holland 1971.
Rogers, H. Jr. 1967 Theory of recursive functions and effective computability. New York: McGraw-Hill 1967.
Scott, D. Invariant Borel Sets. Fund. Math. 56, 117—128 (1964).
Scott, D. Logic with denumerably long formulas and finite strings of quantifiers, The Theory of Models, pp. 329—341. Amsterdam: North-Holland 1965.
Shoenfield, J. R. 1961 The problem of predicativity, Essays on the Foundations of Mathematics, pp. 132—139. Amsterdam: North-Holland 1961.
Shoenfield, J. R. 1967 Mathematical logic. Reading, Mass.: Addison-Wesley 1967.
Simpson, S. G. 1974 Degree Theory on Admissible Ordinals, Generalized Recursion Theory, pp. 165—194. Amsterdam: North-Holland 1974.
Smullyan, R. 1963 A unifying principle in quantification theory. Proc. Nat. Acad. Sci. 49, 828—832 (1963).
Smullyan, R. 1965 A unifying principle in quantification theory, The Theory of Models, edited by Addison, J., Henkin, L. and Tarski, A., pp. 443—434. Amsterdam: North-Holland 1965.
Spector, C. 1955 Recursive wellorderings. J. Symbolic Logic 20, 151—163 (1955).
Spector, C. 1960 Hyperarithmetical quantifiers. Fund. Math. 48, 313—320 (1960).
Spector, C. 1961 Inductively defined sets of natural numbers. In: Infmitistic methods, pp. 97—102. New York: Pergamon 1961.
Suzuki, Y., Wilmers, G. 1973 Nonstandard models for set theory, Proceedings of the Bertrand Russell Memorial Logic Conference, Denmark 1971, pp. 278—314. Leeds 1973.
Svenonius, L. 1965 On the denumerable models of theories with extra predicates, The Theory of Models, pp. 376—389. Amsterdam: North-Holland 1965.
Takeuti, G. 1960 On the recursive functions of ordinal numbers. J. Math. Soc. Japan 12, 119—128 (1960).
Takeuti, G. 1965 Recursive functions and arithmetic functions of ordinal numbers. In: Bar-Hillel, Y. (ed.) Logic, Methodology and Philosophy of Science II (Proceedings of the 1964 Congress) pp. 179—196. Amsterdam: North-Holland 1965.
Tagué, T. 1959 Predicates recursive in a type-2 object and Kleene hierarchies. Comment. Math. Univ. St. Paul 8, 97—117(1959).
Tagué, T. 1964 On the partial recursive functions of ordinal numbers. J. Math. Soc. Japan 16, 1—31 (1964).
Vaught, R. 1973 Descriptive set theory in Lω1ω, Cambridge Summer School in Math. Logic, pp. 574—598 (Lecture Notes in Math., Vol. 337). Berlin-Heidelberg-New York: Springer 1973.
Ville, F. 1974 More on set existence, Generalized Recursion Theory, pp. 195—208. Amsterdam: North- Holland 1974.
Wilmers, G. 1973 An X1-standard model for ZF which is an element of the minimal model, Proceedings of the Bertrand Russell Memorial Logic Conference, Denmark 1971, pp. 215—326. Leeds 1973.


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed.