Higher Recursion Theory

Gerald E. Sacks

doi:10.1017/9781316717301

Aczel, P, Richter, W (1972) Inductive definitions and analogues of large cardinals. In: Conference in Math. Log., London 1970. Springer, Berlin Heidelberg New York, p 1–9

Bailey, C (1984) Beta-degrees for weakly inadmissible sets. Ph.D. Thesis, Harvard University, Cambridge, MA

Barwise, J (1979) Admissible sets and structures. Springer, Berlin Heidelberg New York

Cenzer, D (1976) Monotonic inductive definitions over the continuum J. Symb. Log. 41: 188–198

Chong, CT, Lerman, M (1976) Hyperhypersimple α-r.e. sets. Ann. Math. Log. 9: 1–42

Chong, CT (1979) Generic sets and minimal α-degrees, Trans. Amer. Math. Soc. 254: 157–169

Chong, CT (1984) Techniques of Admissible Recursion Theory. Springer, Berlin Heidelberg New York

Church, A, Kleene, SC (1937) Formal definitions in the theory of ordinal numbers. Fund. Math. 28: 11–21

Dekker, JCE (1954) A theorem on hypersimple sets. Proc. Amer. Math. Soc. 5: 791–796

Devlin, K (1984) Constructibility. Springer, Berlin Heidelberg New York

Driscoll, GC Jr. (1968) Metarecursively enumerable sets and their metadegrees, J. Symb. Log. 33: 389–411

Feferman, S, Spector, C (1962) Incompleteness along paths in progressions of theories. J. Symb. Log. 27: 383–390

Feferman, S (1965) Some applications of the notions of forcing and generic sets. Fund. Math. 56: 325–345

Fenstad, JE (1980) General recursion theory. Springer, Berlin Heidelberg New York

Friedberg, R (1957a) A criterion for completeness of degrees of unsolvability, J. Symb. Log. 22:159–160

Friedberg, R (1957b) Two recursively enumerable sets of incomparable degrees of unsolvability. Proc. Nat. Acad. Sci. USA 43: 236–238

Friedman, SD (1976) Recursion on Inadmissible Ordinals. Ph.D. Thesis, Massachusetts Institute of Technology

Friedman, SD, Sacks, GE (1977) Inadmissible recursion theory, Bull. Math. Soc. 83: 255–256

Friedman, SD (1979) βrecursion theory. Trans. Amer. Math. Soc. 255: 173–200

Friedman, SD (1981a) Uncountable admissibles II: compactness. Israel J. Math. 40: 129–149

Friedman, SD (1981b) Negative solutions to Post's problem. II. Ann. Math. 113: 25–43

Gandy, RO (1960) Proof of Mostowski's conjecture. Bull. Acad. Polon. Sci. Math. 8: 571–575

Gandy, RO (1967) General recursive functionals of finite type and hierarchies of functionals. Ann. Fac. Sci. Univ. Clermont-Ferrand 35: 202–242

Gandy, RO, Sacks, GE (1967) A minimal hyperdegree, Fund. Math. 61: 215–223

Green, J (1974) Σ1 compactness for next admissible sets. J. Symb. Log. 39: 105–116

Griffor, ER (1980) E-Recursively Enumerable Degrees. Ph.D. Thesis, Massachusetts Institute of Technology

Griffor, ER, Normann, D (1982) Effective confinalities and admissibility in Erecursion. Preprint, University of Oslo

Grilliot, T (1969) Selection functions for recursive functionals. Notre Dame Jour. Formal Log. p. 225–234

Harrington, L (1973) Contributions to recursion theory, in higher types. Ph.D. Thesis, (Massachusetts Institute of Technology, Cambridge MA)

Harrington, L, MacQueen, D (1976) Selection in abstract recursion theory, J. Symb. Log. 41: 153–158

Hinman, P (1978) Recursion–Theoretic Hierarchies. Springer, Berlin Heidelberg New York

Homer, S, Sacks, GE (1983) Inverting the half-jump. Trans. Amer. Math. Soc. 278: 317–331

Hoole, T (1982) Abstract extended 2-sections. Ph.D. thesis, Oxford University (Note: the date and title of this reference are approximate.)

Jech, T (1978) Set Theory. Academic Press, New York

Jockusch, C, Simpson, S (1976) A degree-theoretic definition of the ramified analytical hierarchy, Ann. Math. Log. 10: 1–32

Kleene, SC (1943) Recursive predicates and quantifiers, Trans. Amer. Math. Soc. 53: 41–73

Kleene, SC (1952) Introduction to Metamathematics. Van Nostrand, New York.

Kleene, SC (1955a) Arithmetical predicates and function quantifies, Trans. Amer. Math. Soc. 79:312–340

Kleene, SC (1955b) Hierarchies of number-theoretic predicates, Bull. Amer. Math. Soc. 61: 193–213

Kleene, SC (1955c) On the forms of the predicates in the theory of constructive ordinals II, Amer. J. Math. 77: 405–428

Kleene, SC (1959) Recursive functionals and quantifies of finite types I. Trans. Amer. Math. Soc. 91:1–52

Kleene, SC (1963) Recursive functionals and quantifies of finite types II, Trans. Amer. Math. Soc. 108: 106–142

Kreisel, G (1961) Set theoretic methods suggested by the notion of potential totality. In: Infinitistic Methods. Pergamon, Oxford, 325–369

Kreisel, G (1962) The axiom of choice and the class of hyperarithmetic functions, Indag. Math. 24: 307–319

Kreisel, G, Sacks, GE (1963) Metarecursive sets I, II (abstracts), J. Symb. Log. 28: 304–305

Kreisel, G (1965) Model-theoretic invariants: applications to recursive and hyperarithmetic operations. In: Theory of models. North-Holland, Amsterdam, p 190–205

Kreisel, G, Sacks, GE (1965) Metarecursive sets, J. Symb. Log. 30 p 318–338

Kreisel, G (1971) Some reasons for generalizing recursion theory. In: Logic Colloquium ’69. North-Holland Amsterdam, p 139–198

Kripke, S (1964) Transfinite recursion on admissible ordinals I, II (abstracts), J. Symb. Log. 29:161–162

Lachlan, AH (1966) Lower bounds for pairs of recursively enumerable degrees. Proc. London Math. Soc. 16: 537–569

Lachlan, AH (1975) A recursively enumerable degree which will not split over all lesser ones. Ann. Math. Log. 9: 307–365

Levy, A (1963) Transfinite computability (abstract), Notices Amer. Math. Soc. 10: 286

Lerman, M, Sacks, GE (1972) Some minimal pairs of αrecursively enumerable degrees. Ann. Math. Log. 4: 415–42

Lerman, M (1974) Maximal α-r.e. sets. Trans. Amer. Math. Soc. 188: 341–386

Lerman, M (1983) Degrees of unsolvability. Springer, Berlin Heidelberg New York

Louveau, A (1980) A separation theorem for Σ1 1 sets, Trans. Amer. Math. Soc. 260: 363–378

Maass, W (1977a) Minimal pairs and minimal degrees in higher recursion theory. Zeit. Math. Logik 18: 169–186

Maass, W (1977b) Contributions to α- and βrecursion theory. Habilitationsschrift, Universität München

Maass, W (1978a) The uniform regular set theorem in arecursion theory. J. Symb. Log. 43: 270–279

Maass, W (1978b) Inadmissibility, tame r.e. sets and the admissible collapse. Ann. Math. Log. 13: 149–170

Machover, M (1961) The theory of transfinite recursion. Bull. Amer. Math. Soc. 67: 575–578

Machtey, M (1970) Admissible ordinals and intrinsic consistency. J. Symb. Log. 35: 389–400

Macintyre, JM (1968) Contributions to metarecursion theory. Ph.D. Thesis, M.I.T., Cambridge MA

Macintyre, JM (1973) Minimal αrecursion theoretic degrees. J. Symb. Log. 38: 18–28

MacQueen, D (1972) Recursion in finite types. Ph.D. Thesis. M.I.T., Cambridge MA

Moldstad, J (1977) Computations in higher types. Lecture Notes in Math. 574. Springer, Berlin Heidelberg New York

Moschovakis, YN (1966) Many-one degrees of the Ha(x) predicates. Parif. Jour. Math. 18: 329–342

Moschovakis, YN (1980) Descriptive set theory. North-Holland, Amsterdam

Muchnik, AA (1956) On the unsolvability of the problem of reducibility in the theory of algorithms. Dokl. Akad. Nauk SSSR, N.S. 108: 194–197

Normann, D (1974) Imbedding of higher type theories. Preprint Series in Math. 16: Oslo

Normann, D (1975) Degrees of functionals. Preprint Series in Math. 22: Oslo

Normann, D (1978a) Set recursion. In: Generalized recursion theory II. North-Holland, Amsterdam, 303–320

Normann, D (1978b) Recursion in 3 E and a splitting theorem. In: Essays on mathematical and philosophical logic. D. Reidel, Dordrecht, p 275–285

Ohashi, K (1970) On a question of G.E. Sacks. J. Symb. Log. 35: 46–50

Owings, JC Jr. (1969) Π1 1-sets, ω-sets and metacompleteness. J. Symb. Log. 34: 194–204

Platek, R (1966) Foundations of recursion theory, Ph.D. Thesis. Stanford University, Stanford CA

Post, EL (1944) Recursively enumerable sets of positive integers and their decision problems. Bull. Amer. Math. Soc. 50: 284–316

Richter, W (1967) Constructive transfinite number classes. Bull. Amer. Math. Soc. 73: 261–265

Rogers, H Jr. (1967) Theory of recursive functions and effective computability. McGraw-Hill, New York

Sacks, GE (1963a) Recursive enumerability and the jump operator. Trans. Amer. Math. Soc. 108: 223–239

Sacks, GE (1963b) On the degrees less than 0′. Annals of Math. 77: 211–231

Sacks, GE (1964) The recursively enumerable degrees are dense. Ann. of Math. 80: 193–205

Sacks, GE (1966) Post's problem, admissible ordinals and regularity. Trans. Amer. Math. Soc. 124:1–23

Sacks, GE (1969) Measure theoretic uniformity in recursion theory and set theory. Trans. Amer. Math. Soc. 142: 381–420

Sacks, GE (1970) Recursion in objects of finite type. Proc. Internat. Cong. Math. 1: 251–254

Sacks, GE (1971) Forcing with perfect closed sets. In Axiomatic set theory, Proc. Symposia in Pure Math. Amer. Math. Soc. 13: 331–355

Sacks, GE, Simpson, S (1972) The α-finite injury method. Ann. Math. Log. 4: 343–367

Sacks, GE (1974) The 1-section of a type n-object. In: Generalized recursion theory. North-Holland, Amsterdam, p 81–96

Sacks, GE (1976) Countable admissible ordinals and hyperdegrees. Advances in Math. 19: 213–262

Sacks, GE (1977) The k-section of a type n-object. Amer. J. Math. 99: 901–917

Sacks, GE (1980) Post's problem, absoluteness and recursion in finite types. In The Kleene Symposium, North-Holland, Amsterdam, p 201–222

Sacks, GE (1985) Post's problem in Erecursion. In Proceedings of symposia in pure mathematics. Amer. Math. Soc. 42: 177–193

Sacks, GE (1986) On the limits of E-recursive enumerability. Ann. of Pure and Applied Logic 31: 87–120

Sacks, GE, Slaman, TA (1987) Inadmissible forcing. Advances in Math. 66: 1–30

Sacks, GE (199?) Set forcing over E-closed structures (to appear)

Shore, RA (1974) Σ n sets which are ∆ n incomparable (uniformly). J. Symb. Log. 39: 295–304

Shore, RA (1975a) The irregular and non-hyperregular α-r.e. degrees, Israel J. Math. 22: 28–411

Shore, RA (1975b) Splitting an αrecursively enumerable set. Trans. Amer. Math. Soc. 204: 65–78

Shore, RA (1975c) Some more minimal pairs of arecursively enumerable degrees (abstract). Notices Amer. Math. Soc. 22: A524–525

Shore, RA (1976a) The recursively enumerable α-degrees are dense. Ann. Math. Log. 9: 123–155

Shore, RA (1976b) Combining the density and splitting theorem for α-r.e. degrees (abstract). Notices Amer. Math. Soc. 23: A598

Simpson, SG (1971) Admissible Ordinals and Recursion Theory. Ph.D. Thesis, M.I.T., Cambridge MA

Simpson, SG (1974a) Degree theory on admissible ordinals. In Generalized recursion theory, Proceedings of the 1972 Oslo Symposium. North-Holland, Amsterdam, p 165–194

Simpson, SG (1974b) Post's problem for admissible sets. In Generalized recursion theory, Proceedings of the 1972 Oslo Symposium, North-Holland, Amsterdam, p 437–441

Slaman, TA (1981) Aspects of E-Recursion. Ph.D. Thesis. Harvard University, Cambridge MA

Slaman, TA (1983) The extended plus-one hypothesis–a relative consistency result. Nagoya Math. J. 92: 107–120

Slaman, TA (1985a) Reflection and forcing in Erecursion theory. Ann. Pure Appl. Log. 29: 79–106

Slaman, TA (1985b) The Erecursively enumerable degrees are dense. In Proceedings of symposia in pure mathematics. Amer. Math. Soc. 42, 195–213

Soare, RI (1987) Recursively enumerable sets and degrees. Springer, Berlin Heidelberg New York

Spector, C (1955) Recursive well-orderings, J. Symb. Log. 20: 151–163

Spector, C (1956) On degrees of recursive unsolvability, Ann. of Math. 64: 581–592

Spector, C (1959) Hyperarithmetic quantifiers, Fund. Math. 48: 313–320

Stoltenberg-Hansen, V (1977) Finite injury argument in infinite computation theories. Preprint Series in Math. 12: Oslo

Suzuki, Y (1964) A complete classification of the Δ1 2 functions, Bull. Amer. Math. Soc. 70: 246–253

Takeuti, G (1960) On the recursive functions of ordinal numbers. J. Math. Soc. Japan 12: 119–128

Tanaka, H (1968) A basis result for Π1 1-sets of positive measure. Comment. Math. Univ. St. Paul 16: 115–127

Tugué, T (1964) On the partial recursive functions of ordinal numbers. J. Math. Soc. Japan 16: 1–31

vande, Wiele J (1982) Recursive dilators and generalized recursion. In Proceedings of the Herbrand Symposium. North-Holland, Amsterdam, p 325–332

Yang, DP (1984) On the embedding of α-recursive presentable lattices in the a-recursive degrees below 0′. J. Symb. Log. 49: 488–502

Yates, CEM (1966) A minimal pair of recursively enumerable degrees. J. Symb. Log. 31: 159–168

Higher Recursion Theory

This Book has been cited by the following publications. This list is generated based on data provided by Crossref.

Book description

Refine List

Actions for selected content:

Contents

Frontmatter
pp i-iv

Dedication
pp v-vi

Preface to the Series Perspectives in Mathematic Logic
pp vii-viii

Author's Preface
pp ix-xii

Contents
pp xiii-xvi

Part A - Hyperarithmetic Sets
pp 1-2

Chapter I - Constructive Ordinals and Sets
pp 3-21

Chapter II - The Hyperarithmetic Hierarchy
pp 22-51

Chapter III - Predicates of Reals
pp 52-87

Chapter IV - Measure and Forcing
pp 88-112

Part B - Metarecursion
pp 113-114

Chapter V - Metarecursive Enumerability
pp 115-134

Chapter VI - Hyperregularity and Priority
pp 135-148

Part C - α-Recursion
pp 149-150

Chapter VII - Admissibility and Regularity
pp 151-174

Chapter VIII - Priority Arguments
pp 175-203

Chapter IX - Splitting, Density and Beyond
pp 204-230

Part D - E-Recursion
pp 231-232

Chapter X - E-Closed Structures
pp 233-258

Chapter XI - Forcing Computations to Converge
pp 259-282

Chapter XII - Selection and k-Sections
pp 283-308

Chapter XIII - E-Recursively Enumerable Degrees
pp 309-338

Bibliography
pp 339-342

Subject Index
pp 343-344

Metrics

Altmetric attention score

Full text views

Book summary page views

Accessibility standard: Unknown

Why this information is here

Accessibility Information

Higher Recursion Theory

Book description

Refine List

Actions for selected content:

Save Search

Contents

Metrics

Altmetric attention score

Full text views

Book summary page views

Accessibility standard: Unknown

Why this information is here

Accessibility Information