On extensions of models of strong fragments of arithmetic

Kossak, Roman

doi:10.1090/s0002-9939-1990-0984802-2

Cited by 10 publications

(11 citation statements)

References 8 publications

(1 reference statement)

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“…In this section, we state and prove our main theorem on isomorphisms between models of WKL * 0 + ¬IΣ 0 1 . One can view the theorem as a generalization to second-order arithmetic of a result of Kossak's [28, Theorem 3.1] (see [29] or [20] for a correction to the proof) saying that every countable model of BΣ 1 (A) + exp + ¬IΣ 1 (A) has continuum many automorphisms. In fact, our proof is somewhat reminiscent of Kossak's argument, in which one also finds a truth-coding trick that goes back to Smoryński [34, Proof.…”

Section: The Isomorphism Theoremmentioning

confidence: 99%

An isomorphism theorem for models of Weak König's Lemma without primitive recursion

Fiori-Carones¹,

Kołodziejczyk²,

Wong³

et al. 2021

Preprint

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Section: The Isomorphism Theoremmentioning

confidence: 99%

An isomorphism theorem for models of Weak König's Lemma without primitive recursion

Fiori-Carones¹,

Kołodziejczyk²,

Wong³

et al. 2021

Preprint

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“…Loftiness and the -property. This section is concerned with some properties of models that were introduced in [4], [5], and [11]. Results of this section will be used in the proofs of the main results although automorphism groups do not appear here.…”

Section: To What Extent Does Aut(m) Determine Ssy(m)? Th(m)?mentioning

confidence: 99%

“…According to [11], the following lemma is implicit in [5]; it is explicitly proved in [ Proof. We sketch the proof as it will be needed later on.…”

Section: To What Extent Does Aut(m) Determine Ssy(m)? Th(m)?mentioning

confidence: 99%

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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“…The result that any countable K \= BZ" + exp + ~iIS" has 2 No automorphisms is due to Roman Kossak [1990], and many of his ideas are used in the proof of Theorem 3.3 below. Notice that 3.1 follows from 3.2 and 3.3 since, as in the proof of 1.4, by compactness we may take any L >-K with K not cofinal in L and take N = K as given in (ii) of the statement of 3.1.…”

mentioning

confidence: 99%

“…So suppose we are given a countable K \= BZ" + exp + ~i LE", where n > 1. Initially, we follow a modification of the proof in Kossak [1990] that K has continuum-many automorphisms.…”

mentioning

confidence: 99%

Model-theoretic properties characterizing Peano arithmetic

Kaye

1991

J. symb. log.

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Let = {0,1, +,·,<} be the usual first-order language of arithmetic. We show that Peano arithmetic is the least first-order -theory containing IΔ0 + exp such that every complete extension T of it has a countable model K satisfying(i) K has no proper elementary substructures, and(ii) whenever L ≻ K is a countable elementary extension there is and such that .Other model-theoretic conditions similar to (i) and (ii) are also discussed and shown to characterize Peano arithmetic.

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On extensions of models of strong fragments of arithmetic

Cited by 10 publications

References 8 publications

An isomorphism theorem for models of Weak König's Lemma without primitive recursion

An isomorphism theorem for models of Weak König's Lemma without primitive recursion

Automorphism Groups of Countable Arithmetically Saturated Models of Peano Arithmetic

Model-theoretic properties characterizing Peano arithmetic

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