| 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
(författare)
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A New Proof of Tanaka's Theorem
- 2013
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Ingår i: New Studies in Weak Arithmetics. Patrick Cégielski, Charalampos Cornaros, and Costas Dimitracopoulos (eds.). CSLI Lectures Notes, No.211. - 9781575867236 ; , s. 93-102
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Konferensbidrag (refereegranskat)abstract
- We give a new proof of a theorem of Kazuyuki Tanaka, which states that every countable nonstandard model of WKL_0 has a self-embedding onto a proper initial segment of itself.
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| 4. |
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| 5. |
- Enayat, Ali, 1959, et al.
(författare)
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Axiomatizations of Peano Arithmetic: a truth-theoretic view
- 2023
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Ingår i: Journal of Symbolic Logic. - : Cambridge University Press (CUP). - 0022-4812 .- 1943-5886. ; 88:4, s. 1526-1555
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Tidskriftsartikel (refereegranskat)abstract
- We study the family of axiomatizations of Peano arithemtic, both locally and globally, using the lens provided by axiomatic truth theory.
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| 6. |
- Enayat, Ali, 1959, et al.
(författare)
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Categoricity-like properties in the first-order realm
- 2024
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- By classical results of Dedekind and Zermelo, second order logic imposes categoricity features on Peano Arithmetic and Zermelo-Fraenkel set theory. However, we have known since Skolem’s anticategoricity theorems that the first order formulations of Peano Arithmetic and Zermelo-Fraenkel set theory (i.e., PA and ZF) are not categorical. Here we investigate various categoricity-like properties (including tightness, solidity, and internal categoricity) that are exhibited by a distinguished class of first order theories that include PA and ZF, with the aim of understanding what is special about canonical foundational first order theories.
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| 7. |
- Enayat, Ali, 1959
(författare)
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Condensable models of set theory
- 2022
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Ingår i: Archive for mathematical logic. - : Springer Science and Business Media LLC. - 0933-5846 .- 1432-0665. ; 61:3-4, s. 299-315
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Tidskriftsartikel (refereegranskat)abstract
- We study models M of set theory that are "condensable", in the sense that there is an "ordinal" o of M such that the rank initial segment of M determined by o is both isomorphic to M, and also an elementary submodel of M for infinitary formulae in the well-founded part of M. We prove, assuming a modest set theoretic hypothesis, that there are condensable models M of ZFC such that every definable element of M is in the well-founded part of M. We also provide a structural characterization of condensable models of ZF.
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| 8. |
- Enayat, Ali, 1959
(författare)
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Curious satisfaction classes
- 2023
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We present two new constructions of satisfaction/truth classes over models of PA (Peano Arithmetic) that provide a foil to the fact that the existence of a disjunctively correct full truth class over a model M of PA implies that Con(PA) holds in M.
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| 9. |
- 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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| 10. |
- Enayat, Ali, 1959, et al.
(författare)
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Incompleteness of boundedly axiomatizable theories
- 2023
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Our main result (Theorem A) shows the incompleteness of any consistent sequential theory T formulated in a finite language such that T is axiomatized by a collection of sentences of bounded quantifier-alternation-depth. Our proof employs an appropriate reduction mechanism to rule out the possibility of completeness by simply invoking Tarski's Undefinability of Truth theorem. We also use the proof strategy of Theorem A to obtain other incompleteness results (as in Theorems A+; B and B+).
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