| 1. |
- Bahrami, Saeideh, et al.
(författare)
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Fixed points of self-embeddings of models of arithmetic
- 2018
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Ingår i: Annals of Pure and Applied Logic. - : Elsevier BV. - 0168-0072. ; 169:6, s. 487-513
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Tidskriftsartikel (refereegranskat)abstract
- We investigate the structure of fixed point sets of self-embeddings of models of arithmetic. In particular, given a countable nonstandard model M of a modest fragment of Peano arithimetic, we provide complete characterizations of (a) the initial segments of M that can be realized as the longest initial segment of fixed points of a nontrivial self-embedding of M onto a proper initial segment of M; and (b) the initial segments of M that can be realized as the fixed point set of some nontrivial self-embedding of M onto a proper initial segment of M. Moreover, we demonstrate the the standard cut is strong in M iff there is a self-embedding of M onto a proper initial segment of itself that moves every element that is not definable in M by an existential formula.
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| 2. |
- Blanck, Rasmus, 1982, et al.
(författare)
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Marginalia on a theorem of Woodin
- 2017
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Ingår i: Journal of Symbolic Logic. - : Cambridge University Press (CUP). - 0022-4812 .- 1943-5886. ; 82:1, s. 359-374
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Tidskriftsartikel (refereegranskat)abstract
- Let ⟨Wn : n ∈ ω⟩ be a canonical enumeration of recursively enumerable sets, and suppose T is a recursively enumerable extension of PA (Peano Arithmetic) in the same language. Woodin (2011) showed that there exists an index e∈ω (that depends on T) with the property that if M is a countable model of T and for some M-finite set s, M satisfies We⊆s, then M has an end extension N that satisfies T + We=s. Here we generalize Woodin’s theorem to all recursively enumerable extensions T of the fragment IΣ1 of PA, and remove the countability restriction on M when T extends PA. We also derive model-theoretic consequences of a classic fixed-point construction of Kripke (1962) and compare them with Woodin’s theorem.
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| 3. |
- Enayat, Ali, 1959, et al.
(författare)
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Feferman's Forays into the Foundations of Category Theory
- 2017
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Ingår i: Feferman on Foundations / Jaeger, Gerhard, Sieg, Wilfried (Eds.). - Cham : Springer International Publishing. - 2211-2758. - 9783319633329 ; , s. 315-346
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Bokkapitel (refereegranskat)abstract
- The foundations of category theory has been a source of many perplexities ever since the groundbreaking 1945-introduction of the subject by Eilenberg and Mac Lane; e.g., how is one to avoid Russell-like paradoxes and yet have access to objects that motivate the study in the first place, such as the category of all groups, or the category of all topological spaces? Solomon Feferman has grappled with such perplexities for over 45 years, as witnessed by his six papers on the subject during the period 1969-2013. Our focus in this paper is on two important, yet quite different set-theoretical systems proposed by Feferman for the implementation of category theory: the ZF-style system ZFC/S and the NFU-style system S*; where NFU is Jensen's urelemente-modication of Quine's New Foundations system NF of set theory.
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| 4. |
- Enayat, Ali, 1959, et al.
(författare)
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Iterated ultrapowers for the masses
- 2018
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Ingår i: Archive for mathematical logic. - : Springer Science and Business Media LLC. - 0933-5846 .- 1432-0665. ; 57:5-6, August 2018, s. 557-576
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Tidskriftsartikel (refereegranskat)abstract
- We present a novel, perspicuous framework for building iterated ultrapowers. Furthermore, our framework naturally lends itself to the construction of a certain type of order indiscernibles, here dubbed tight indiscernibles, which are shown to provide smooth proofs of several results in general model theory.
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| 5. |
- Enayat, Ali, 1959, et al.
(författare)
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Largest initial segments pointwise fixed by automorphisms of models of set theory
- 2018
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Ingår i: Archive for Mathematical Logic. - : Springer Science and Business Media LLC. - 0933-5846 .- 1432-0665. ; 57:1-2, s. 91-139
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Tidskriftsartikel (refereegranskat)abstract
- Given a model M of set theory, and a nontrivial automorphism j of M, let I_fix(j) be the submodel of M whose universe consists of elements m of M such that j(x)=x for every x in the transitive closure of m (where the transitive closure of m is computed within M. Here we study models of the form I_fix(j) and characterize their first order theory.
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| 6. |
- Enayat, Ali, 1959, et al.
(författare)
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New Constructions of Satisfaction Classes
- 2015
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Ingår i: Unifying the Philosophy of Truth / Theodora Achourioti, Henri Galinon José Martínez Fernández, Kentaro Fujimoto Editors. - Dordrecht : Springer. - 2214-9775. - 9789401796736 ; , s. 321-335
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Bokkapitel (refereegranskat)abstract
- We use model-theoretic ideas to present a perspicuous and versatile method of constructing full satisfaction classes on models of Peano arithmetic. We also comment on the ramifications of our work on issues related to conservativity and interpretability.
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| 7. |
- Enayat, Ali, 1959, et al.
(författare)
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Truth, Disjunction, and Induction
- 2019
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Ingår i: Archive for mathematical logic. - : Springer Science and Business Media LLC. - 0933-5846 .- 1432-0665. ; 58:5-6, s. 753-766
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Tidskriftsartikel (refereegranskat)abstract
- By a well-known result of Kotlarski et al. (1981), first-order Peano arithmetic PA can be conservatively extended to the theory CT−[PA] of a truth predicate satisfying compositional axioms, i.e., axioms stating that the truth predicate is correct on atomic formulae and commutes with all the propositional connectives and quantifiers. This result motivates the general question of determining natural axioms concerning the truth predicate that can be added to CT−[PA] while maintaining conservativity over PA . Our main result shows that conservativity fails even for the extension of CT−[PA] obtained by the seemingly weak axiom of disjunctive correctness DC that asserts that the truth predicate commutes with disjunctions of arbitrary finite size. In particular, CT−[PA]+DC implies Con(PA) . Our main result states that the theory CT−[PA]+DC coincides with the theory CT0[PA] obtained by adding Δ0 -induction in the language with the truth predicate. This result strengthens earlier work by Kotlarski (1986) and Cieśliński (2010). For our proof we develop a new general form of Visser’s theorem on non-existence of infinite descending chains of truth definitions and prove it by reduction to (Löb’s version of) Gödel’s second incompleteness theorem, rather than by using the Visser–Yablo paradox, as in Visser’s original proof (1989).
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| 8. |
- Enayat, Ali, 1959, et al.
(författare)
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Unifying the model theory of first-order and second-order arithmetic via WKL*_0
- 2017
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Ingår i: Annals of Pure and Applied Logic. - : Elsevier BV. - 0168-0072. ; 30:6, s. 1247-1283
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Tidskriftsartikel (refereegranskat)abstract
- We develop machinery to make the Arithmetized Completeness Theorem more effective in the study of many models of IΔ0+BΣ1+exp, including all countable ones, by passing on to the conservative extension View the MathML source of IΔ0+BΣ1+exp. Our detailed study of the model theory of View the MathML source leads to the simplification and improvement of many results in the model theory of Peano arithmetic and its fragments pertaining to the construction of various types of end extensions and initial segments.
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| 9. |
- Enayat, Ali, 1959
(författare)
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Variations on a Visserian Theme
- 2016
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Ingår i: Liber Amicorum Alberti : a tribute to Albert Visser / Jan van Eijck, Rosalie Iemhoff and Joost J. Joosten (eds.). - London : College Publications. - 9781848902046 ; , s. 99-110
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Bokkapitel (refereegranskat)
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| 10. |
- Enayat, Ali, 1959, et al.
(författare)
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ZFC proves that the class of ordinals is not weakly compact for definable classes
- 2018
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Ingår i: Journal of Symbolic Logic. - : Cambridge University Press (CUP). - 0022-4812 .- 1943-5886. ; 83:1, s. 146-164
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Tidskriftsartikel (refereegranskat)abstract
- © 2018 The Association for Symbolic Logic. In ZFC, the class Ord of ordinals is easily seen to satisfy the definable version of strong inaccessibility. Here we explore deeper ZFC-verifiable combinatorial properties of Ord, as indicated in Theorems A & B below. Note that Theorem A shows the unexpected result that Ord is never definably weakly compact in any model of ZFC. Theorem A. LetMbe any model of ZFC. (1) The definable tree property fails in M: There is an M-definable Ord-tree with no M-definable cofinal branch. (2) The definable partition property fails in M: There is an M-definable 2-coloring f: [X] 2 → 2 for someM-definable proper class X such that no M-definable proper classs is monochromatic for f. (3) The definable compactness property for L∞, ω fails in M: There is a definable theory Γ in the logic L∞,ω (in the sense ofM) of size Ord such that every set-sized subtheory of Γ is satisfiable in M, but there is no M-definable model of Γ. Theorem B. The definable Ord principle holds in a model M of ZFC iff M carries an M-definable global well-ordering. Theorems A and Babove can be recast as theoremschemes in ZFC, or as asserting that a single statement in the language of class theory holds in all 'spartan' models of GB (Godel-Bernays class theory); where a spartan model of GB is any structure of the form (M,DM), where M | = ZF and DM is the family of M-definable classes. Theorem C gauges the complexity of the collection GB spa of (Godel-numbers of) sentences that hold in a ll spartan models of GB. Theorem C. GB spa is Π 1 1 -complete.
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