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| 2. |
- 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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| 3. |
- 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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| 4. |
- 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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| 5. |
- 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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| 6. |
- 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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| 7. |
- Enayat, Ali, 1959
(författare)
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Indiscernibles and satisfaction classes in arithmetic
- 2024
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Ingår i: Archive for mathematical logic. - 0933-5846 .- 1432-0665. ; 63:5-6, s. 655-677
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Tidskriftsartikel (refereegranskat)abstract
- We investigate the theory PAI (Peano Arithmetic with Indiscernibles). Models of PAI are of the form (M, I), where M is a model of PA, I is an unbounded set of order indiscernibles over M, and (M, I) satisfies the extended induction scheme for formulae mentioning I. Our main results are Theorems A and B below. Theorem A. Let M be a nonstandard model of PA of any cardinality. M has an expansion to a model of PAI iff M has an inductive partial satisfaction class. Theorem A yields the following corollary, which provides a new characterization of countable recursively saturated models of PA: Corollary. A countable model M of PA is recursively saturated iff M has an expansion to a model of PAI. Theorem B. There is a sentence s in the language obtained by adding a unary predicate I(x) to the language of arithmetic such that given any nonstandard model M of PA of any cardinality, M has an expansion to a model of PAI + s iff M has a inductive full satisfaction class.
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| 8. |
- Enayat, Ali, 1959, et al.
(författare)
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Initial self-embeddings of models of set theory
- 2021
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Ingår i: Journal of Symbolic Logic. - : Cambridge University Press (CUP). - 0022-4812 .- 1943-5886. ; 86:4, s. 1584-1611
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Tidskriftsartikel (refereegranskat)abstract
- By a classical theorem of Harvey Friedman (1973), every countable nonstandard model Mof a suciently strong fragment of ZF has a proper rank-initial self-embedding j, i.e., j is a self-embedding of M such that j[M] (M, and the ordinal rank of each member of j[M] is less than the ordinal rank of each element of Mn j[M]. Here we investigate the larger family of proper initial-embeddings j of models M of fragments of set theory, where the image of j is a transitive submodel of M. Our results include the following three theorems. In what follows, ZF is ZF without the power set axiom; WO is the axiom stating that every set can be well-ordered; WF(M) is the well-founded part of M; and 1 1DC is the full scheme of dependent choice of length.
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| 9. |
- Enayat, Ali, 1959
(författare)
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Models of Set Theory: Extensions and Dead-ends
- 2024
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- This paper is a contribution to the study of extensions of arbitrary models of ZF (Zermelo-Fraenkel set theory), with no regard to countability or well-foundedness of the models involved. We present some new constructions of certain types of extensions, and also establish the existence of models of ZF that cannot be properly end extended.
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| 10. |
- Enayat, Ali, 1959, et al.
(författare)
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On Effectively Indiscernible Projective Sets and the Leibniz-Mycielski Axiom
- 2021
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Ingår i: Mathematics. - : MDPI AG. - 2227-7390. ; 9:14
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Tidskriftsartikel (refereegranskat)abstract
- Examples of effectively indiscernible projective sets of real numbers in various models of set theory are presented. We prove that it is true, in Miller and Laver generic extensions of the constructible universe, that there exists a lightface Π21 equivalence relation on the set of all nonconstructible reals, having exactly two equivalence classes, neither one of which is ordinal definable, and therefore the classes are OD-indiscernible. A similar but somewhat weaker result is obtained for Silver extensions. The other main result is that for any n, starting with 2, the existence of a pair of countable disjoint OD-indiscernible sets, whose associated equivalence relation belongs to lightface Πn1, does not imply the existence of such a pair with the associated relation in Σn1 or in a lower class.
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