Results for 'Recursively saturated model'

995 found
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  1.  37
    Automorphisms of Countable Short Recursively Saturated Models of PA.Erez Shochat - 2008 - Notre Dame Journal of Formal Logic 49 (4):345-360.
    A model of Peano Arithmetic is short recursively saturated if it realizes all its bounded finitely realized recursive types. Short recursively saturated models of $\PA$ are exactly the elementary initial segments of recursively saturated models of $\PA$. In this paper, we survey and prove results on short recursively saturated models of $\PA$ and their automorphisms. In particular, we investigate a certain subgroup of the automorphism group of such models. This subgroup, denoted (...)
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  2.  13
    Recursively saturated models generated by indiscernibles.James H. Schmerl - 1985 - Notre Dame Journal of Formal Logic 26 (2):99-105.
  3.  30
    Automorphisms of Countable Recursively Saturated Models of PA: Open Subgroups and Invariant Cuts.Henryk Kotlarski & Bozena Piekart - 1995 - Mathematical Logic Quarterly 41 (1):138-142.
    Let M be a countable recursively saturated model of PA and H an open subgroup of G = Aut. We prove that I = sup {b ∈ M : ∀u < bfu = u and J = inf{b ∈ MH} may be invariant, i. e. fixed by all automorphisms of M.
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  4.  26
    Automorphisms of recursively saturated models of arithmetic.Richard Kaye, Roman Kossak & Henryk Kotlarski - 1991 - Annals of Pure and Applied Logic 55 (1):67-99.
    We give an examination of the automorphism group Aut of a countable recursively saturated model M of PA. The main result is a characterisation of strong elementary initial segments of M as the initial segments consisting of fixed points of automorphisms of M. As a corollary we prove that, for any consistent completion T of PA, there are recursively saturated countable models M1, M2 of T, such that Aut[ncong]Aut, as topological groups with a natural topology. (...)
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  5.  44
    Automorphisms of Countable Recursively Saturated Models of PA: A Survey.Henryk Kotlarski - 1995 - Notre Dame Journal of Formal Logic 36 (4):505-518.
    We give a survey of automorphisms of countable recursively saturated models of Peano Arithmetic.
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  6.  35
    Four Problems Concerning Recursively Saturated Models of Arithmetic.Roman Kossak - 1995 - Notre Dame Journal of Formal Logic 36 (4):519-530.
    The paper presents four open problems concerning recursively saturated models of Peano Arithmetic. One problems concerns a possible converse to Tarski's undefinability of truth theorem. The other concern elementary cuts in countable recursively saturated models, extending automorphisms of countable recursively saturated models, and Jonsson models of PA. Some partial answers are given.
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  7.  16
    Elementary extensions of recursively saturated models of arithmetic.C. Smoryński - 1981 - Notre Dame Journal of Formal Logic 22 (3):193-203.
  8.  12
    On maximal subgroups of the automorphism group of a countable recursively saturated model of PA.Roman Kossak, Henryk Kotlarski & James H. Schmerl - 1993 - Annals of Pure and Applied Logic 65 (2):125-148.
    We show that the stabilizer of an element a of a countable recursively saturated model of arithmetic M is a maximal subgroup of Aut iff the type of a is selective. This is a point of departure for a more detailed study of the relationship between pointwise and setwise stabilizers of certain subsets of M and the types of elements in those subsets. We also show that a complete type of PA is 2-indiscernible iff it is minimal (...)
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  9.  18
    A Galois correspondence for countable short recursively saturated models of PA.Erez Shochat - 2010 - Mathematical Logic Quarterly 56 (3):228-238.
    In this paper we investigate the properties of automorphism groups of countable short recursively saturated models of arithmetic. In particular, we show that Kaye's Theorem concerning the closed normal subgroups of automorphism groups of countable recursively saturated models of arithmetic applies to automorphism groups of countable short recursively saturated models as well. That is, the closed normal subgroups of the automorphism group of a countable short recursively saturated model of PA are (...)
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  10.  23
    Ω1-like recursively saturated models of Presburger's arithmetic.Victor Harnik - 1986 - Journal of Symbolic Logic 51 (2):421-429.
  11.  15
    Decoding in the automorphism group of a recursively saturated model of arithmetic.Ermek Nurkhaidarov - 2015 - Mathematical Logic Quarterly 61 (3):179-188.
    The main result of this paper partially answers a question raised in about the existence of countable just recursively saturated models of Peano Arithmetic with non‐isomorphic automorphism groups. We show the existence of infinitely many countable just recursively saturated models of Peano Arithmetic such that their automorphism groups are not topologically isomorphic. We also discuss maximal open subgroups of the automorphism group of a countable arithmetically saturated model of in a very good interstice.
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  12.  25
    The ω-like recursively saturated models of arithmetic.Roman Kossak - 1991 - Bulletin of the Section of Logic 20 (3/4):109-109.
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  13.  42
    Recursively saturated nonstandard models of arithmetic.C. Smoryński - 1981 - Journal of Symbolic Logic 46 (2):259-286.
  14.  20
    A Note on Real Subsets of A Recursively Saturated Model.Athanassios Tzouvaras - 1991 - Mathematical Logic Quarterly 37 (13‐16):207-216.
  15.  35
    On two questions concerning the automorphism groups of countable recursively saturated models of PA.Roman Kossak & Nicholas Bamber - 1996 - Archive for Mathematical Logic 36 (1):73-79.
  16.  17
    On closed elementary cuts in recursively saturated models of Peano arithmetic.Bożena Piekart - 1993 - Notre Dame Journal of Formal Logic 34 (2):223-230.
  17.  24
    Recursively saturated $\omega_1$-like models of arithmetic.Roman Kossak - 1985 - Notre Dame Journal of Formal Logic 26 (4):413-422.
  18.  45
    Recursively saturated nonstandard models of arithmetic; addendum.C. Smoryński - 1982 - Journal of Symbolic Logic 47 (3):493-494.
  19.  27
    A Note on Real Subsets of A Recursively Saturated Model.Athanassios Tzouvaras - 1991 - Mathematical Logic Quarterly 37 (13-16):207-216.
  20.  13
    The Recursively Saturated Part of Models of Peano Arithmetic.Henryk Kotlarski - 1986 - Mathematical Logic Quarterly 32 (19‐24):365-370.
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  21.  20
    A note on initial segment constructions in recursively saturated models of arithmetic.C. Smoryński - 1982 - Notre Dame Journal of Formal Logic 23 (4):393-408.
  22.  26
    The Recursively Saturated Part of Models of Peano Arithmetic.Henryk Kotlarski - 1986 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 32 (19-24):365-370.
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  23.  38
    Toward model theory through recursive saturation.John Stewart Schlipf - 1978 - Journal of Symbolic Logic 43 (2):183-206.
  24.  25
    Arithmetically Saturated Models of Arithmetic.Roman Kossak & James H. Schmerl - 1995 - Notre Dame Journal of Formal Logic 36 (4):531-546.
    The paper presents an outline of the general theory of countable arithmetically saturated models of PA and some of its applications. We consider questions concerning the automorphism group of a countable recursively saturated model of PA. We prove new results concerning fixed point sets, open subgroups, and the cofinality of the automorphism group. We also prove that the standard system of a countable arithmetically saturated model of PA is determined by the lattice of its (...)
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  25.  19
    An Introduction to Recursively Saturated and Resplendent Models.Jon Barwise & John Schlipf - 1982 - Journal of Symbolic Logic 47 (2):440-440.
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  26.  35
    Jon Barwise and John Schlipf. On recursively saturated models of arithmetic. Model theory and algebra, A memorial tribute to Abraham Robinson, edited by D. H. Saracino and V. B. Weispfenning, Lecture notes in mathematics, vol. 498, Springer-Verlag, Berlin, Heidelberg, and New York, 1975, pp. 42–55. - Patrick Cegielski, Kenneth McAloon, and George Wilmers. Modèles récursivement saturés de l'addition et de la multiplication des entiers naturels. Logic Colloquium '80, Papers intended for the European summer meeting of the Association for Symbolic Logic, edited by D. van Dalen, D. Lascar, and T. J. Smiley, Studies in logic and the foundations of mathematics, vol. 108, North-Holland Publishing Company, Amsterdam, New York, and London, 1982, pp. 57–68. - Julia F. Knight. Theories whose resplendent models are homogeneous. Israel journal of mathematics, vol. 42 , pp. 151–161. - Julia Knight and Mark Nadel. Expansions of models and Turing degrees. The journal of symbolic logic, vol. 47 , pp. 58. [REVIEW]J. -P. Ressayre - 1987 - Journal of Symbolic Logic 52 (1):279-284.
  27.  47
    An introduction to recursively saturated and resplendent models.Jon Barwise & John Schlipf - 1976 - Journal of Symbolic Logic 41 (2):531-536.
  28.  17
    A generalization of the Keisler-Morley theorem to recursively saturated ordered structures.Shahram Mohsenipour - 2007 - Mathematical Logic Quarterly 53 (3):289-294.
    We prove a model theoretic generalization of an extension of the Keisler-Morley theorem for countable recursively saturated models of theories having a K-like model, where K is an inaccessible cardinal.
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  29.  73
    Nonstandard characterizations of recursive saturation and resplendency.Stuart T. Smith - 1987 - Journal of Symbolic Logic 52 (3):842-863.
    We prove results about nonstandard formulas in models of Peano arithmetic which complement those of Kotlarski, Krajewski, and Lachlan in [KKL] and [L]. This enables us to characterize both recursive saturation and resplendency in terms of statements about nonstandard sentences. Specifically, a model M of PA is recursively saturated iff M is nonstandard and M-logic is consistent.M is resplendent iff M is nonstandard, M-logic is consistent, and every sentence φ which is consistent in M-logic is contained in (...)
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  30.  68
    On interstices of countable arithmetically saturated models of Peano arithmetic.Nicholas Bamber & Henryk Kotlarski - 1997 - Mathematical Logic Quarterly 43 (4):525-540.
    We give some information about the action of Aut on M, where M is a countable arithmetically saturated model of Peano Arithmetic. We concentrate on analogues of moving gaps and covering gaps inside M.
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  31.  29
    Automorphisms of Saturated and Boundedly Saturated Models of Arithmetic.Ermek S. Nurkhaidarov & Erez Shochat - 2011 - Notre Dame Journal of Formal Logic 52 (3):315-329.
    We discuss automorphisms of saturated models of PA and boundedly saturated models of PA. We show that Smoryński's Lemma and Kaye's Theorem are not only true for countable recursively saturated models of PA but also true for all boundedly saturated models of PA with slight modifications.
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  32.  21
    Some highly saturated models of Peano arithmetic.James H. Schmerl - 2002 - Journal of Symbolic Logic 67 (4):1265-1273.
    Some highly saturated models of Peano Arithmetic are constructed in this paper, which consists of two independent sections. In § 1 we answer a question raised in [10] by constructing some highly saturated, rather classless models of PA. A question raised in [7], [3], ]4] is answered in §2, where highly saturated, nonstandard universes having no bad cuts are constructed.Highly saturated, rather classless models of Peano Arithmetic were constructed in [10]. The main result proved there is (...)
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  33.  26
    Jon Barwise and John Schlipf. An introduction to recursively saturated and resplendent models. The journal of symbolic logic, vol. 41 , pp. 531–536.Julia F. Knight - 1982 - Journal of Symbolic Logic 47 (2):440.
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  34.  7
    Automorphism Groups of Countable Arithmetically Saturated Models of Peano Arithmetic.James H. Schmerl - 2015 - Journal of Symbolic Logic 80 (4):1411-1434.
    If${\cal M},{\cal N}$are countable, arithmetically saturated models of Peano Arithmetic and${\rm{Aut}}\left( {\cal M} \right) \cong {\rm{Aut}}\left( {\cal N} \right)$, then the Turing-jumps of${\rm{Th}}\left( {\cal M} \right)$and${\rm{Th}}\left( {\cal N} \right)$are recursively equivalent.
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  35.  8
    James H. Schmerl. Peano models with many generic classes. Pacific Journal of Mathematics, vol. 43 (1973), pp. 523–536. - James H. Schmerl. Correction to: “Peano models with many generic classes”. Pacific Journal of Mathematics, vol. 92 (1981), no. 1, pp. 195–198. - James H. Schmerl. Recursively saturated, rather classless models of Peano arithmetic. Logic Year 1979–80. Recursively saturated, rather classless models of Peano arithmetic. Logic Year 1979–80 (Proceedings, Seminars, and Conferences in Mathematical Logic, University of Connecticut, Storrs, Connecticut, 1979/80). edited by M. Lerman, J. H. Schmerl, and R. I. Soare, Lecture Notes in Mathematics, vol. 859. Springer, Berlin, pp. 268–282. - James H. Schmerl. Recursively saturatedmodels generated by indiscernibles. Notre Dane Journal of Formal Logic, vol. 26 (1985), no. 1, pp. 99–105. - James H. Schmerl. Large resplendent models generated by indiscernibles. The Journal of Symbolic Logic, vol. 54 (1989), no. 4, pp. 1382–1388. - Jam. [REVIEW]Roman Kossak - 2009 - Bulletin of Symbolic Logic 15 (2):222-227.
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  36.  11
    Review: Jon Barwise, John Schlipf, An Introduction to Recursively Saturated and Resplendent Models. [REVIEW]Julia F. Knight - 1982 - Journal of Symbolic Logic 47 (2):440-440.
  37.  10
    James H. Schmerl. Peano models with many generic classes. Pacific Journal of Mathematics, vol. 43 (1973), pp. 523–536. - James H. Schmerl. Correction to: “Peano models with many generic classes”. Pacific Journal of Mathematics, vol. 92 (1981), no. 1, pp. 195–198. - James H. Schmerl. Recursively saturated, rather classless models of Peano arithmetic. Logic Year 1979–80. Recursively saturated, rather classless models of Peano arithmetic. Logic Year 1979–80 (Proceedings, Seminars, and Conferences in Mathematical Logic, University of Connecticut, Storrs, Connecticut, 1979/80). edited by M. Lerman, J. H. Schmerl, and R. I. Soare, Lecture Notes in Mathematics, vol. 859. Springer, Berlin, pp. 268–282. - James H. Schmerl. Recursively saturatedmodels generated by indiscernibles. Notre Dane Journal of Formal Logic, vol. 26 (1985), no. 1, pp. 99–105. - James H. Schmerl. Large resplendent models generated by indiscernibles. The Journal of Symbolic Logic, vol. 54 (1989), no. 4, pp. 1382–1388. - Jam. [REVIEW]Roman Kossak - 2009 - Bulletin of Symbolic Logic 15 (2):222-227.
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  38.  15
    Automorphisms of Models of True Arithmetic: Subgroups which Extend to a Maximal Subgroup Uniquely.Henryk Kotlarski & Bożena Piekart - 1994 - Mathematical Logic Quarterly 40 (1):95-102.
    We show that if M is a countable recursively saturated model of True Arithmetic, then G = Aut has nonmaximal open subgroups with unique extension to a maximal subgroup of Aut.
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  39. Resplendent models and $${\Sigma_1^1}$$ -definability with an oracle.Andrey Bovykin - 2008 - Archive for Mathematical Logic 47 (6):607-623.
    In this article we find some sufficient and some necessary ${\Sigma^1_1}$ -conditions with oracles for a model to be resplendent or chronically resplendent. The main tool of our proofs is internal arguments, that is analogues of classical theorems and model-theoretic constructions conducted inside a model of first-order Peano Arithmetic: arithmetised back-and-forth constructions and versions of the arithmetised completeness theorem, namely constructions of recursively saturated and resplendent models from the point of view of a model (...)
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  40.  53
    Models with the ω-property.Roman Kossak - 1989 - Journal of Symbolic Logic 54 (1):177-189.
    A model M of PA has the omega-property if it has a subset of order type omega that is coded in an elementary end extension of M. All countable recursively saturated models have the omega-property, but there are also models with the omega-property that are not recursively saturated. The papers is devoted to the study of structural properties of such models.
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  41.  12
    Automorphisms of Models of True Arithmetic: More on Subgroups which Extend to a Maximal One Uniquely.Henryk Kotlarski & Bożena Piekart - 2000 - Mathematical Logic Quarterly 46 (1):111-120.
    Continuing the earlier research in [14] we give some more information about nonmaximal open subgroups of G = Aut with unique maximal extension, where ℳ is a countable recursively saturated model of True Arithmetic.
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  42.  40
    Modèles saturés et modèles engendrés Par Des indiscernables.Benoît Mariou - 2001 - Journal of Symbolic Logic 66 (1):325-348.
    In the early eighties, answering a question of A. Macintyre, J. H. Schmerl ([13]) proved that every countable recursively saturated structure, equipped with a function β encoding the finite functions, is the β-closure of an infinite indiscernible sequence. This result implies that every countably saturated structure, in a countable but not necessarily recursive language, is an Ehrenfeucht-Mostowski model, by which we mean that the structure expands, in a countable language, to the Skolem hull of an infinite (...)
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  43.  77
    The complexity of classification problems for models of arithmetic.Samuel Coskey & Roman Kossak - 2010 - Bulletin of Symbolic Logic 16 (3):345-358.
    We observe that the classification problem for countable models of arithmetic is Borel complete. On the other hand, the classification problems for finitely generated models of arithmetic and for recursively saturated models of arithmetic are Borel; we investigate the precise complexity of each of these. Finally, we show that the classification problem for pairs of recursively saturated models and for automorphisms of a fixed recursively saturated model are Borel complete.
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  44.  34
    A new spectrum of recursive models using an amalgamation construction.Uri Andrews - 2011 - Journal of Symbolic Logic 76 (3):883 - 896.
    We employ an infinite-signature Hrushovski amalgamation construction to yield two results in Recursive Model Theory. The first result, that there exists a strongly minimal theory whose only recursively presentable models are the prime and saturated models, adds a new spectrum to the list of known possible spectra. The second result, that there exists a strongly minimal theory in a finite language whose only recursively presentable model is saturated, gives the second non-trivial example of a (...)
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  45.  11
    Condensable models of set theory.Ali Enayat - 2022 - Archive for Mathematical Logic 61 (3):299-315.
    A model \ of ZF is said to be condensable if \\prec _{\mathbb {L}_{{\mathcal {M}}}} {\mathcal {M}}\) for some “ordinal” \, where \:=,\in )^{{\mathcal {M}}}\) and \ is the set of formulae of the infinitary logic \ that appear in the well-founded part of \. The work of Barwise and Schlipf in the 1970s revealed the fact that every countable recursively saturated model of ZF is cofinally condensable \prec _{\mathbb {L}_{{\mathcal {M}}}}{\mathcal {M}}\) for an unbounded collection (...)
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  46.  59
    Models of weak theories of truth.Mateusz Łełyk & Bartosz Wcisło - 2017 - Archive for Mathematical Logic 56 (5):453-474.
    In the following paper we propose a model-theoretical way of comparing the “strength” of various truth theories which are conservative over $$ PA $$. Let $${\mathfrak {Th}}$$ denote the class of models of $$ PA $$ which admit an expansion to a model of theory $${ Th}$$. We show (combining some well known results and original ideas) that $$\begin{aligned} {{\mathfrak {PA}}}\supset {\mathfrak {TB}}\supset {{\mathfrak {RS}}}\supset {\mathfrak {UTB}}\supseteq \mathfrak {CT^-}, \end{aligned}$$ where $${\mathfrak {PA}}$$ denotes simply the class of all (...)
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  47.  15
    Closed Normal Subgroups.James H. Schmerl - 2001 - Mathematical Logic Quarterly 47 (4):489-492.
    Let ℳ be a countable, recursively saturated model of Peano Arithmetic, and let Aut be its automorphism group considered as a topological group with the pointwise stabilizers of finite sets being the basic open subgroups. Kaye proved that the closed normal subgroups are precisely the obvious ones, namely the stabilizers of invariant cuts. A proof of Kaye's theorem is given here which, although based on his proof, is different enough to yield consequences not obtainable from Kaye's proof.
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  48.  34
    Models of PT- with Internal Induction for Total Formulae.Cezary Cieslinski, Bartosz Wcisło & Mateusz Łełyk - 2017 - Review of Symbolic Logic 10 (1):187-202.
    We show that a typed compositional theory of positive truth with internal induction for total formulae (denoted by PT tot ) is not semantically conservative over Peano arithmetic. In addition, we observe that the class of models of PA expandable to models of PT tot contains every recursively saturated model of arithmetic. Our results point to a gap in the philosophical project of describing the use of the truth predicate in model-theoretic contexts.
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  49.  44
    Transplendent Models: Expansions Omitting a Type.Fredrik Engström & Richard W. Kaye - 2012 - Notre Dame Journal of Formal Logic 53 (3):413-428.
    We expand the notion of resplendency to theories of the kind T + p", where T is a fi rst-order theory and p" expresses that the type p is omitted. We investigate two di erent formulations and prove necessary and sucient conditions for countable recursively saturated models of PA. Some of the results in this paper can be found in one of the author's doctoral thesis [3].
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  50.  14
    Moving Intersticial Gaps.James H. Schmerl - 2002 - Mathematical Logic Quarterly 48 (2):283-296.
    In a countable, recursively saturated model of Peano Arithmetic, an interstice is a maximal convex set which does not contain any definable elements. The interstices are partitioned into intersticial gaps in a way that generalizes the partition of the unbounded interstice into gaps. Continuing work of Bamber and Kotlarski [1], we investigate extensions of Kotlarski's Moving Gaps Lemma to the moving of intersticial gaps.
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