An unpublished theorem of Solovay, revisited
A definable pair of disjoint non-OD sets of reals (hence, indiscernible sets) exists in the Sacks and $E_0$-large generic extensions of the constructible universe $L$.
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arXiv subjects
Ali Enayat
Publications and source records attributed to Ali Enayat.
Truth and Feasible Reducibility
Let $\mathcal{T}$ be any of the three canonical truth theories $\textsf{CT}^-$ (Compositional truth without extra induction), $\textsf{FS}^-$ (Friedman--Sheard truth without extra induction), and $\textsf{KF}^-$ (Kripke--Feferman truth without extra induction), where the base theory of $\mathcal{T}$ is $\textsf{PA}$ (Peano arithmetic). We show that $\mathcal{T}$ is \textit{feasibly reducible to} $\textsf{PA}$, i.e., there is a polynomial time computable function $f$ such that for any proof $π$ of an arithmetical sentence $ϕ$ in $\mathcal{T}$, $f(π)$ is a proof of $ϕ$ in $\textsf{PA}$. In particular, $\mathcal{T}$ has at most polynomial speed-up over $\textsf{PA}$, in sharp contrast to the situation for $\mathcal{T}[\textsf{B}]$ for \textit{finitely axiomatizable} base theories $\textsf{B}$.
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Truth, Disjunction, and Induction
By a well-known result of Kotlarski, Krajewski, and Lachlan (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 results 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 $CT_0[PA]$ obtained by adding $Δ_0$-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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Fixed Points of Self-embeddings of Models of Arithmetic
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 segement 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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ZFC proves that the class of ordinals is not weakly compact for definable classes
We prove that the class of all ordinals Ord is not weakly compact with respect to definable classes. Specifically, in any model of ZFC, the definable tree property fails for Ord, in that there is a definable Ord tree with no definable cofinal branch; the definable partition property fails, in that there is a definable 2-coloring of pairs from a certain definable proper class, with no definable homogeneous proper class; and the definable compactness property fails for $\mathcal{L}_{\infty,ω}$, in that there is a definable theory in this logic all of whose set-sized subtheories are satisfiable, but which has no definable proper class model. In addition, we prove that the definable diamond principle $\Diamond_{\rm Ord}$ holds if and only if there is a definable well-ordering of the universe. And we prove that the common theory of all spartan models of Gödel-Bernays set theory, those having only definable classes, is $Π^1_1$-complete.
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Iterated Ultrapowers for the Masses
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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Variations on a Visserian Theme
A first order theory T is said to be "tight" if for any two deductively closed extensions U and V of T (both of which are formulated in the language of T), U and V are bi-interpretable iff U = V. By a theorem of Visser, PA (Peano Arithmetic) is tight. Here we show that Z_2 (second order arithmetic), ZF (Zermelo-Fraenkel set theory), and KM (Kelley-Morse theory of classes) are also tight theories.
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Largest initial segments pointwise fixed by automorphisms of models of set theory
Given a model $\mathcal{M}$ of set theory, and a nontrivial automorphism $j$ of $\mathcal{M}$, let $\mathcal{I}_{\mathrm{fix}}(j)$ be the submodel of $\mathcal{M}$ whose universe consists of elements $m$ of $\mathcal{M}$ such that $j(x)=x$ for every $x$ in the transitive closure of $m$ (where the transitive closure of $m$ is computed within $\mathcal{M}$). Here we study the class $\mathcal{C}$ of structures of the form $\mathcal{I}_{\mathrm{fix}}(j)$, where the ambient model $\mathcal{M}$ satisfies a frugal yet robust fragment of $\mathrm{ZFC}$ known as $\mathrm{MOST}$, and $j(m)=m$ whenever $m$ is a finite ordinal in the sense of $\mathcal{M}$. We show that every structure in $\mathcal{C}$ satisfies $\mathrm{MOST}+Δ_0^\mathcal{P}\textrm{-Collection}$. We also show that the following countable structures are in $\mathcal{C}$: (a) transitive models of $\mathrm{MOST}+Δ_0^\mathcal{P}\textrm{-Collection}$, (b) recursively saturated models of $\mathrm{MOST}+Δ_0^\mathcal{P}\textrm{-Collection}$, (c) models of $\mathrm{ZFC}$. It follows from (b) that the theory of $\mathcal{C}$ is precisely $\mathrm{MOST+Δ}_{0}^{\mathcal{P}}$-Collection. We conclude by proving a refinement of a result due to Amir Togha.
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Feferman's Forays into the Foundations of Category Theory
This paper is primarily concerned with assessing a set-theoretical system, $S^*$, for the foundations of category theory suggested by Solomon Feferman. $S^*$ is an extension of NFU, and may be seen as an attempt to accommodate unrestricted categories such as the category of all groups (without any small/large restrictions), while still obtaining the benefits of ZFC on part of the domain. A substantial part of the paper is devoted to establishing an improved upper bound on the consistency strength of $S^*$. The assessment of $S^*$ as a foundation of category theory is framed by the following general desiderata (R) and (S). (R) asks for the unrestricted existence of the category of all groups, the category of all categories, the category of all functors between two categories, etc., along with natural implementability of ordinary mathematics and category theory. (S) asks for a certain relative distinction between large and small sets, and the requirement that they both enjoy the full benefits of the $\mathrm{ZFC}$ axioms. $S^*$ satisfies (R) simply because it is an extension of NFU. By means of a recursive construction utilizing the notion of strongly cantorian sets, we argue that it also satisfies (S). Moreover, this construction yields a lower bound on the consistency strength of $S^*$. We also exhibit a basic positive result for category theory internal to NFU that provides motivation for studying NFU-based foundations of category theory.
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