The Structure of Models of Peano Arithmetic

Kossak, Roman; Schmerl, James H.

doi:10.1093/acprof:oso/9780198568278.001.0001

Cited by 90 publications

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“…(2) =⇒ (1): This part of the proof relies on the technology for constructing models with a prescribed substructure lattice as presented in [15,Chap. 4.5].…”

Section: 4])mentioning

confidence: 99%

“…Notation and terminology used here will generally follow [15]. The reader should refer to [15] for insufficiently explained notions. The proof of Theorem 4 relies on a number of results that are proved or stated in [19].…”

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confidence: 99%

“…Smoryński actually proved more (see [15,Theorem 8.4.2]): If M is a countable recursively saturated model and I ⊆ M is an exponentially closed cut, then there is f ∈ Aut(M) such that I = I fix (f ) = {x ∈ M : f (y) = y for all y < x}.…”

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confidence: 99%

“…If X is a set of subsets of ω and X ⊆ ω, then X is enumerated by X if X = {(X) i : i < ω}. The next proposition collects together some well known properties of countable, recursively saturated models most of which can be found in various places in [15].…”

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Automorphism Groups of Countable Arithmetically Saturated Models of Peano Arithmetic

Schmerl¹

2015

J. symb. log.

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Abstract. If M, N are countable, arithmetically saturated models of Peano Arithmetic and Aut(M) ∼ = Aut(N ), then the Turingjumps of Th(M) and Th(N ) are recursively equivalent.Since 1991, when the question Are there countable, recursively saturated models M, N of PA such that Aut(M) ∼ = Aut(N ) (as abstract groups)? appeared in [8], it has been of interest to determine to what extent (the isomorphism type of) the group Aut(M) of all automorphisms of a countable, recursively saturated model M of Peano Arithmetic determines (the isomorphism type of) M. It was proved in [8] that whenever both M and N are countable, recursively saturated models of PA and exactly one of them is arithmetically saturated, then Aut(M) and Aut(N ) are not isomorphic as topological groups. In 1994, Lascar [16] proved that countable, arithmetically saturated models of PA have the small index property, and that result then implied that Aut(M) ∼ = Aut(N ) as abstract groups. This gave the first positive answer to the above question. A neater way, in which the use of the small index property is masked, that automorphism groups distinguish those models that are arithmetically saturated from the other countable recursively saturated ones was obtained the next year in [13, Coro. 3.9] (or see [15, Th. 9.3.10]): If M is countable and recursively saturated, then M is arithmetically saturated iff the cofinality of Aut(M) is uncountable. Finally, we mention that Kaye's Theorem [7] (see §1) characterizing the closed normal subgroups of Aut(M), which appeared in the same volume [9] as did Lascar's Theorem, yields that whenever M, N are countable, arithmetically saturated models and M is a model of True Arithmetic (TA) while N is not, then Aut(M) ∼ = Aut(N ).

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“…(2) =⇒ (1): This part of the proof relies on the technology for constructing models with a prescribed substructure lattice as presented in [15,Chap. 4.5].…”

Section: 4])mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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confidence: 99%

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Automorphism Groups of Countable Arithmetically Saturated Models of Peano Arithmetic

Schmerl¹

2015

J. symb. log.

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“…For example: by a theorem of Smoryński ([23], [14 Our Theorem 3.4 is an analogue of (1); while Theorems 5.5, 5.6, and 5.15 provide analogues of (2). More generally, our results can be viewed as contributing to the project initiated in [3] of investigating the extent to which core results about automorphisms of model of arithmetic can be extended to the set-theoretic realm.…”

Section: Introductionmentioning

confidence: 99%

Largest initial segments pointwise fixed by automorphisms of models of set theory

2017

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Generic cuts in models of arithmetic

Kaye

2008

Mathematical Logic Qtrly

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We present some general results concerning the topological space of cuts of a countable model of arithmetic given by a particular indicator Y .The notion of 'indicator' is defined in a novel way, without initially specifying what property is indicated and is used to define a topological space of cuts of the model. Various familiar properties of cuts (strength, regularity, saturation, coding properties) are investigated in this sense, and several results are given stating whether or not the set of cuts having the property is comeagre.A new notion of 'generic cut' is introduced and investigated and it is shown in the case of countable arithmetically saturated models M PA that generic cuts exist, indeed the set of generic cuts is comeagre in the sense of Baire, and furthermore that two generic cuts within the same 'small interval' of the model are conjugate by an automorphism of the model.The paper concludes by outlining some applications to constructions of cuts satisfying properties incompatible with genericity, and discussing in model-theoretic terms those properties for which there is an indicator Y .

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The Structure of Models of Peano Arithmetic

Cited by 90 publications

References 0 publications

Automorphism Groups of Countable Arithmetically Saturated Models of Peano Arithmetic

Automorphism Groups of Countable Arithmetically Saturated Models of Peano Arithmetic

Largest initial segments pointwise fixed by automorphisms of models of set theory

Generic cuts in models of arithmetic

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