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On extensions of models of strong fragments of arithmetic

Roman Kossak
<span title="1990-01-01">1990</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/a64sw4kcsveajhudnssdwrwvze" style="color: black;">Proceedings of the American Mathematical Society</a> </i> &nbsp;
From a partial solution to this problem we deduce a generalization of the theorem of Smorynski and Stavi on cofinal extensions of recursively saturated models of arithmetic.  ...  We consider the problem of not almost semiregularity of models of /X" + -"5S"+i .  ...  Introduction By a strong fragment of arithmetic we mean every theory in the language of Peano Arithmetic extending FL0 + i?X, + exp.  ... 
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On Extensions of Models of Strong Fragments of Arithmetic

Roman Kossak
<span title="">1990</span> <i title="JSTOR"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/a64sw4kcsveajhudnssdwrwvze" style="color: black;">Proceedings of the American Mathematical Society</a> </i> &nbsp;
From a partial solution to this problem we deduce a generalization of the theorem of Smorynski and Stavi on cofinal extensions of recursively saturated models of arithmetic.  ...  We consider the problem of not almost semiregularity of models of /X" + -"5S"+i .  ...  Introduction By a strong fragment of arithmetic we mean every theory in the language of Peano Arithmetic extending FL0 + i?X, + exp.  ... 
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Page 4727 of Mathematical Reviews Vol. , Issue 92i [page]

<span title="">1992</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
of fragments of arithmetic.  ...  Symbolic Logic 51 (1986), no. 4, 1022-1028; MR 88b:03049] gave a syntactic characterization of the formulas preserved by cofinal extensions of fragments of arithmetic.  ... 
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Page 5251 of Mathematical Reviews Vol. , Issue 91J [page]

<span title="">1991</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Athanassios Tzouvaras (GR-THESS) 91j:03050 03C62 03HI5 Kossak, Roman (PL-PAN) A correction to: “On extensions of models of strong fragments of arithmetic” [Proc. Amer. Math.  ...  The main theorem is: Every model of ZFC has a cofinal conservative extension which possesses a min- imal cofinal extension. The proof uses the Boolean ultrapower construction.  ... 
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Partial collapses of the Σ1 complexity hierarchy in models for fragments of bounded arithmetic

Zofia Adamowicz, Leszek Aleksander Kołodziejczyk
<span title="">2007</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/bnojym2hjzcgnpa4wixi2axhnq" style="color: black;">Annals of Pure and Applied Logic</a> </i> &nbsp;
For any n, we construct a model of T n 2 + ¬exp in which each ∃s b n+1 formula is equivalent to an ∃ b n formula.  ...  We want to consider fragments of diagrams of models, so given a model M, we work not just with the usual bounded arithmetic language L 2 , but also with its extension L(M) obtained by adding a constant  ...  M is a b n+1maximal model of T n 2 containing an element a such that the standard iterations of # on a are cofinal in M.  ... 
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Page 3376 of Mathematical Reviews Vol. , Issue 88g [page]

<span title="">1988</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Then 7 has a model which is not recursively saturated but every proper, simple cofinal extension of which is N,-saturated. (3) If « >; is regular, then every x-saturated model of PA has a x-saturated,  ...  proper, simple cofinal extension which is not x* saturated.  ... 
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Automorphisms of models of bounded arithmetic

Ali Enayat
<span title="">2006</span> <i title="Institute of Mathematics, Polish Academy of Sciences"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/yeobluctzze7rfwekxs7sx2ayy" style="color: black;">Fundamenta Mathematicae</a> </i> &nbsp;
We establish the following model theoretic characterization of the fragment I∆ 0 +Exp+BΣ 1 of Peano arithmetic in terms of fixed points of automorphisms of models of bounded arithmetic (the fragment I∆  ...  The following two conditions are equivalent for a countable model M of the language of arithmetic: (a) M satisfies I∆ 0 + BΣ 1 + Exp.  ...  schema with restricted induction), and Z 2 (full second order arithmetic) in terms of automorphisms of models of bounded arithmetic (the fragment I∆ 0 of P A).  ... 
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Sub-arithmetical ultrapowers: a survey

Thomas G. McLaughlin
<span title="">1990</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/bnojym2hjzcgnpa4wixi2axhnq" style="color: black;">Annals of Pure and Applied Logic</a> </i> &nbsp;
By Gaifman's theorem on cofinal extensions [9, Theorem 31, COF,,,(X' ) is an elementary extension of X'.  ...  Can an isomorphic copy of a A: ultrapower sit cofinally inside an Z-structure that satisfies a fragment of true arithmetic of level ~?j(A~+,) or higher?  ...  (3) Can a AZ ultrapower be cofinally (though of course not Aft-elementarily and cofinally) embedded in a model of %?g(Ajl+,) Arithmetic?  ... 
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Peano arithmetic and hyper-Ramsey logic

James H. Schmerl
<span title="1986-02-01">1986</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/w3g32txdvneltemssag5nwfxcy" style="color: black;">Transactions of the American Mathematical Society</a> </i> &nbsp;
A by-product and ingredient of the proof is, for example, the existence of a model of CA having the form (¿V, C\ass(J^)).  ...  St"), which is Peano arithmetic in the context of JP^", has the same first-order consequences as II¡,-CA0.  ...  that: "It would be interesting to find other fragments of second-order arithmetic with nonstandard models in the intended (Cantorian) semantics."  ... 
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<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180725093705/https://www.ams.org/journals/tran/1986-296-02/S0002-9947-1986-0846594-0/S0002-9947-1986-0846594-0.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/ae/7f/ae7fe5d990f76cd92f0c08067d9e86d6b5e5aa1e.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0002-9947-1986-0846594-0"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Peano Arithmetic and Hyper-Ramsey Logic

James H. Schmerl
<span title="">1986</span> <i title="JSTOR"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/w3g32txdvneltemssag5nwfxcy" style="color: black;">Transactions of the American Mathematical Society</a> </i> &nbsp;
A by-product and ingredient of the proof is, for example, the existence of a model of CA having the form (¿V, C\ass(J^)).  ...  St"), which is Peano arithmetic in the context of JP^", has the same first-order consequences as II¡,-CA0.  ...  that: "It would be interesting to find other fragments of second-order arithmetic with nonstandard models in the intended (Cantorian) semantics."  ... 
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Page 2001 of Mathematical Reviews Vol. , Issue 96d [page]

<span title="">1996</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
This paper studies the question of provability of lower bounds on the computational complexity of explicitly given Boolean func- tions in weak fragments of Peano arithmetic.  ...  For abelian torsion groups, the cofinality of their Scott spectrum can be determined from the well-known extension by Barwise and Eklof to torsion groups of arbitrary cardinality of Ulm’s theorem for countable  ... 
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Page 45 of Mathematical Reviews Vol. , Issue 90A [page]

<span title="">1990</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
(C) is provable in a fragment ACAp + RT of second-order arithmetic with the infinite Ramsey theorem, and it is not provable in ACAo alone.  ...  The next stronger analogue (C) states that the same formula has an w-model in which P is cofinal if and only if it has a finite stretchable model in which P consists of at least n special indiscernibles  ... 
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Truth definitions without exponentiation and the Σ1 collection scheme

Zofia Adamowicz, Leszek Aleksander Kołodziejczyk, J. Paris
<span title="">2012</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/4l7nckgxmbcgvj5vxsioq6qwyq" style="color: black;">Journal of Symbolic Logic (JSL)</a> </i> &nbsp;
We prove that: • if there is a model of IΔ0 + ¬exp with cofinal Σ1-definable elements and a Σ1 truth definition for Σ1 sentences, then IΔ0 + ¬exp + ¬BΣ1 is consistent, • there is a model of IΔ0 + Ω1 +  ...  The latter result is obtained by constructing a model with a recursive truth-preserving translation of Σ1 sentences into boolean combinations of sentences.  ...  We assume that the reader is familiar with basic notions and results concerning fragments of Peano Arithmetic and bounded arithmetic (all relevant information can be found in [HP93] ).  ... 
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A standard model of Peano arithmetic with no conservative elementary extension

Ali Enayat
<span title="">2008</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/bnojym2hjzcgnpa4wixi2axhnq" style="color: black;">Annals of Pure and Applied Logic</a> </i> &nbsp;
numbers such that the expansion N A := (N, A) A∈A of the standard model N := (ω, +, ×) of Peano arithmetic has no conservative elementary extension, i.e., for any elementary extension there is a subset  ...  The principal result of this paper answers a long-standing question in the model theory of arithmetic [R. Kossak, J.  ...  I am also indebted to Jim Schmerl for invaluable comments on earlier drafts of this paper (many of which are noted throughout the paper), and to the anonymous referee, especially for advice on structuring  ... 
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Page 820 of Mathematical Reviews Vol. , Issue 81C [page]

<span title="">1981</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Mostowski that simple fragments of primitive recursive arithmetic fail to have recursive models other than the standard one.  ...  Finally, he applies a consequence of work of Jensen to the effect that if m is a sentence of L(Q,,Q,) and @ has a model in some generic extension of the universe in which GCH holds, then @ has a model  ... 
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