Ali Enayat
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Ali Enayat has a total of 55 co-authors. Here are the top co-authors by number of shared publications:
- Zachiri McKenzie
Department of Mathematics University of Chester Chester, UK
9 shared publications - James Cummings
Department of Sociology, University of York, UK
8 shared publications - Nam Trang8 shared publications
- Peter Dybjer
Chalmers University of Technology
8 shared publications - Øystein Linnebo
University of Oslo, Oslo, Norway
8 shared publications - Antonio Montalbán
Department of Mathematics, University of California, Berkeley, USA
7 shared publications
Latest Publications
- Open Access
- 24 March 2026
- Published by Cambridge University Press (CUP) in The Journal of Symbolic Logic
- p. 1-38
MODELS OF SET THEORY: EXTENSIONS AND DEAD-ENDS
This article is a contribution to the study of extensions of arbitrary models of $\mathsf {ZF}$ (Zermelo–Fraenkel set theory), with no regard to countability or well-foundedness of the models involved. Our main results include the theorems below; in Theorems A and B, ${\mathcal {N}}$ is said to be a conservative elementary extension of $\mathcal {M}$ if $\mathcal { N}$ elementarily extends $\mathcal {M}$ , and the intersection of every $ {\mathcal {N}}$ -definable set with the universe of $\mathcal {M}$ is $\mathcal {M} $ -definable (parameters allowed). In Theorem B, $\mathsf {ZFC}$ is the result of augmenting $\mathsf {ZF}$ with the axiom of choice. Theorem A. Every model $\mathcal {M}$ of $\mathsf {ZF}+\exists p\left ( \mathrm {V}=\mathrm {HOD}(p)\right ) $ has a conservative elementary extension ${\mathcal {N}}$ that contains an ordinal above all of the ordinals of $\mathcal {M}$ . Theorem B. If ${\mathcal {N}}$ is a conservative elementary extension of a model $\mathcal {M}$ of $ \mathsf {ZFC}$ , and ${\mathcal {N}}$ has the same natural numbers as $\mathcal {M}$ , then $\mathcal {M}$ is cofinal in ${\mathcal {N}}$ . Theorem C. Every consistent extension of $ \mathrm {ZF}$ has a model $\mathcal {M}$ of power $\aleph _{1}$ such that $\mathcal {M}$ has no proper end extension to a model of $\mathsf {ZF}$ .- 24 March 2026
- Published by Cambridge University Press (CUP) in The Journal of Symbolic Logic
- p. 1-38
Open AccessMODELS OF SET THEORY: EXTENSIONS AND DEAD-ENDS
This article is a contribution to the study of extensions of arbitrary models of $\mathsf {ZF}$ (Zermelo–Fraenkel set theory), with no regard to countability or well-foundedness of the models involved. Our main results include the theorems below; in Theorems A and B, ${\mathcal {N}}$ is said to be a conservative elementary extension of $\mathcal {M}$ if $\mathcal { N}$ elementarily extends $\mathcal {M}$ , and the intersection of every $ {\mathcal {N}}$ -definable set with the universe of $\mathcal {M}$ is $\mathcal {M} $ -definable (parameters allowed). In Theorem B, $\mathsf {ZFC}$ is the result of augmenting $\mathsf {ZF}$ with the axiom of choice. Theorem A. Every model $\mathcal {M}$ of $\mathsf {ZF}+\exists p\left ( \mathrm {V}=\mathrm {HOD}(p)\right ) $ has a conservative elementary extension ${\mathcal {N}}$ that contains an ordinal above all of the ordinals of $\mathcal {M}$ . Theorem B. If ${\mathcal {N}}$ is a conservative elementary extension of a model $\mathcal {M}$ of $ \mathsf {ZFC}$ , and ${\mathcal {N}}$ has the same natural numbers as $\mathcal {M}$ , then $\mathcal {M}$ is cofinal in ${\mathcal {N}}$ . Theorem C. Every consistent extension of $ \mathrm {ZF}$ has a model $\mathcal {M}$ of power $\aleph _{1}$ such that $\mathcal {M}$ has no proper end extension to a model of $\mathsf {ZF}$ . - 13 December 2024
- Published by Cambridge University Press (CUP) in The Review of Symbolic Logic
- Vol. 18 (2) , 545-562
SATISFACTION CLASSES WITH APPROXIMATE DISJUNCTIVE CORRECTNESS
The seminal Krajewski–Kotlarski–Lachlan theorem (1981) states that every countable recursively saturated model of $\mathsf {PA}$ (Peano arithmetic) carries a full satisfaction class. This result implies that the compositional theory of truth over $\mathsf {PA}$ commonly known as $\mathsf {CT}^{-}[\mathsf {PA}]$ is conservative over $\mathsf {PA}$ . In contrast, Pakhomov and Enayat (2019) showed that the addition of the so-called axiom of disjunctive correctness (that asserts that a finite disjunction is true iff one of its disjuncts is true) to $\mathsf {CT}^{-}[\mathsf {PA}]$ axiomatizes the theory of truth $\mathsf {CT}_{0}[\mathsf {PA}]$ that was shown by Wcisło and Łełyk (2017) to be nonconservative over $\mathsf {PA}$ . The main result of this paper (Theorem 3.12) provides a foil to the Pakhomov–Enayat theorem by constructing full satisfaction classes over arbitrary countable recursively saturated models of $\mathsf {PA}$ that satisfy arbitrarily large approximations of disjunctive correctness. This shows that in the Pakhomov–Enayat theorem the assumption of disjunctive correctness cannot be replaced with any of its approximations.- 13 December 2024
- Published by Cambridge University Press (CUP) in The Review of Symbolic Logic
- Vol. 18 (2) , 545-562
SATISFACTION CLASSES WITH APPROXIMATE DISJUNCTIVE CORRECTNESS
The seminal Krajewski–Kotlarski–Lachlan theorem (1981) states that every countable recursively saturated model of $\mathsf {PA}$ (Peano arithmetic) carries a full satisfaction class. This result implies that the compositional theory of truth over $\mathsf {PA}$ commonly known as $\mathsf {CT}^{-}[\mathsf {PA}]$ is conservative over $\mathsf {PA}$ . In contrast, Pakhomov and Enayat (2019) showed that the addition of the so-called axiom of disjunctive correctness (that asserts that a finite disjunction is true iff one of its disjuncts is true) to $\mathsf {CT}^{-}[\mathsf {PA}]$ axiomatizes the theory of truth $\mathsf {CT}_{0}[\mathsf {PA}]$ that was shown by Wcisło and Łełyk (2017) to be nonconservative over $\mathsf {PA}$ . The main result of this paper (Theorem 3.12) provides a foil to the Pakhomov–Enayat theorem by constructing full satisfaction classes over arbitrary countable recursively saturated models of $\mathsf {PA}$ that satisfy arbitrarily large approximations of disjunctive correctness. This shows that in the Pakhomov–Enayat theorem the assumption of disjunctive correctness cannot be replaced with any of its approximations.- 20 September 2024
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 152 (11) , 4923-4932
Incompleteness of boundedly axiomatizable theories
Our main result (Theorem A) shows the incompleteness of any consistent sequential theory T T formulated in a finite language such that T 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.
- 20 September 2024
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 152 (11) , 4923-4932
Incompleteness of boundedly axiomatizable theories
Our main result (Theorem A) shows the incompleteness of any consistent sequential theory T T formulated in a finite language such that T 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.
- Open Access
- 10 September 2024
- Published by Firenze University Press in Journal for the Philosophy of Mathematics
- Vol. 1, 63-98
Categoricity-like Properties in the First Order Realm
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 anti-categoricity 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.
- 10 September 2024
- Published by Firenze University Press in Journal for the Philosophy of Mathematics
- Vol. 1, 63-98
Open AccessCategoricity-like Properties in the First Order Realm
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 anti-categoricity 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.
- Open Access
- 1 September 2024
- Published by Cambridge University Press (CUP) in The Bulletin of Symbolic Logic
- Vol. 30 (3) , f1-f4
BSL volume 30 issue 3 Cover and Front matter
Symbolic Logic, an international organization for furthering research in logic and the exchange of ideas among mathematicians, computer scientists, linguists, and others interested in this fi eld.
- 1 September 2024
- Published by Cambridge University Press (CUP) in The Bulletin of Symbolic Logic
- Vol. 30 (3) , f1-f4
Open AccessBSL volume 30 issue 3 Cover and Front matter
Symbolic Logic, an international organization for furthering research in logic and the exchange of ideas among mathematicians, computer scientists, linguists, and others interested in this fi eld.
- Open Access
- 1 June 2024
- Published by Cambridge University Press (CUP) in The Bulletin of Symbolic Logic
- Vol. 30 (2) , f1-f4
BSL volume 30 issue 2 Cover and Front matter
Symbolic Logic, an international organization for furthering research in logic and the exchange of ideas among mathematicians, computer scientists, linguists, and others interested in this fi eld.
- 1 June 2024
- Published by Cambridge University Press (CUP) in The Bulletin of Symbolic Logic
- Vol. 30 (2) , f1-f4
Open AccessBSL volume 30 issue 2 Cover and Front matter
Symbolic Logic, an international organization for furthering research in logic and the exchange of ideas among mathematicians, computer scientists, linguists, and others interested in this fi eld.
- Open Access
- 28 March 2024
- Published by Springer Nature in Archive for Mathematical Logic
- Vol. 63 (5-6) , 655-677
Indiscernibles and satisfaction classes in arithmetic
We investigate the theory Peano Arithmetic with Indiscernibles ( $$\textrm{PAI}$$ ). Models of $$\textrm{PAI}$$ are of the form $$({\mathcal {M}},I)$$ , where $${\mathcal {M}}$$ is a model of $$\textrm{PA}$$ , I is an unbounded set of order indiscernibles over $${\mathcal {M}}$$ , and $$({\mathcal {M}},I)$$ satisfies the extended induction scheme for formulae mentioning I. Our main results are Theorems A and B following. Theorem A.Let $${\mathcal {M}}$$ be a nonstandard model of $$\textrm{PA}$$ of any cardinality. $$\mathcal {M }$$ has an expansion to a model of $$\textrm{PAI}$$ iff $$ {\mathcal {M}}$$ has an inductive partial satisfaction class. Theorem A yields the following corollary, which provides a new characterization of countable recursively saturated models of $$\textrm{PA}$$ : Corollary.A countable model $${\mathcal {M}}$$ of $$\textrm{PA}$$ is recursively saturated iff $${\mathcal {M}}$$ has an expansion to a model of $$\textrm{PAI}$$ . Theorem B.There is a sentence $$\alpha $$ in the language obtained by adding a unary predicateI(x) to the language of arithmetic such that given any nonstandard model $${\mathcal {M}}$$ of $$\textrm{PA}$$ of any cardinality, $${\mathcal {M}}$$ has an expansion to a model of $$\text {PAI}+\alpha $$ iff $${\mathcal {M}}$$ has a inductive full satisfaction class.- 28 March 2024
- Published by Springer Nature in Archive for Mathematical Logic
- Vol. 63 (5-6) , 655-677
Open AccessIndiscernibles and satisfaction classes in arithmetic
We investigate the theory Peano Arithmetic with Indiscernibles ( $$\textrm{PAI}$$ ). Models of $$\textrm{PAI}$$ are of the form $$({\mathcal {M}},I)$$ , where $${\mathcal {M}}$$ is a model of $$\textrm{PA}$$ , I is an unbounded set of order indiscernibles over $${\mathcal {M}}$$ , and $$({\mathcal {M}},I)$$ satisfies the extended induction scheme for formulae mentioning I. Our main results are Theorems A and B following. Theorem A.Let $${\mathcal {M}}$$ be a nonstandard model of $$\textrm{PA}$$ of any cardinality. $$\mathcal {M }$$ has an expansion to a model of $$\textrm{PAI}$$ iff $$ {\mathcal {M}}$$ has an inductive partial satisfaction class. Theorem A yields the following corollary, which provides a new characterization of countable recursively saturated models of $$\textrm{PA}$$ : Corollary.A countable model $${\mathcal {M}}$$ of $$\textrm{PA}$$ is recursively saturated iff $${\mathcal {M}}$$ has an expansion to a model of $$\textrm{PAI}$$ . Theorem B.There is a sentence $$\alpha $$ in the language obtained by adding a unary predicateI(x) to the language of arithmetic such that given any nonstandard model $${\mathcal {M}}$$ of $$\textrm{PA}$$ of any cardinality, $${\mathcal {M}}$$ has an expansion to a model of $$\text {PAI}+\alpha $$ iff $${\mathcal {M}}$$ has a inductive full satisfaction class. - Open Access
- 1 March 2024
- Published by Cambridge University Press (CUP) in The Bulletin of Symbolic Logic
- Vol. 30 (1) , f1-f3
BSL volume 30 issue 1 Cover and Front matter
Symbolic Logic, an international organization for furthering research in logic and the exchange of ideas among mathematicians, computer scientists, linguists, and others interested in this fi eld.
- 1 March 2024
- Published by Cambridge University Press (CUP) in The Bulletin of Symbolic Logic
- Vol. 30 (1) , f1-f3
Open AccessBSL volume 30 issue 1 Cover and Front matter
Symbolic Logic, an international organization for furthering research in logic and the exchange of ideas among mathematicians, computer scientists, linguists, and others interested in this fi eld.
- Open Access
Incompleteness of boundedly axiomatizable theories
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+).
Open AccessIncompleteness of boundedly axiomatizable theories
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+).
- Open Access
- 1 September 2023
- Published by Cambridge University Press (CUP) in The Bulletin of Symbolic Logic
- Vol. 29 (3) , f1-f3
BSL volume 29 issue 3 Cover and Front matter
Symbolic Logic, an international organization for furthering research in logic and the exchange of ideas among mathematicians, computer scientists, linguists, and others interested in this fi eld.
- 1 September 2023
- Published by Cambridge University Press (CUP) in The Bulletin of Symbolic Logic
- Vol. 29 (3) , f1-f3
Open AccessBSL volume 29 issue 3 Cover and Front matter
Symbolic Logic, an international organization for furthering research in logic and the exchange of ideas among mathematicians, computer scientists, linguists, and others interested in this fi eld.