arXiv subjects

Zachiri McKenzie

Publications and source records attributed to Zachiri McKenzie.

15 recordsLinked to original sources

arXiv2026

On a slight weakening of Kripke-Platek Set Theory

The weak set theory $\mathsf{ReR}$ is obtained from Kripke-Platek Set Theory ($\mathsf{KP}$) by replacing the bounded collection scheme with the bounded replacement scheme. We show that $\mathsf{ReR}$ proves $\mathsf{TCo}$, which asserts that every set is contained in a transitive set. This is used to show that the theories obtained by adding the negation of the axiom of infinity to $\mathsf{ReR}$ and $\mathsf{KP}$ have the same consequences. Our proof of $\mathsf{TCo}$ relies on the availability of a fragment of class foundation in $\mathsf{ReR}$. To demonstrate the necessity of this reliance, even in the presence of infinity, we build a model of a significant fragment of $\mathsf{ZF}$ that includes bounded separation and collection, infinity, powerset, regularity and the axiom of choice, in which $\mathsf{TCo}$ fails.

math.LO↗

arXiv2025

The set-theoretic Kaufmann-Clote question

Let $\mathsf{M}$ be the set theory obtained from $\mathsf{ZF}$ by removing the collection scheme, restricting separation to $Δ_0$-formulae and adding an axiom asserting that every set is contained in a transitive set. Let $Π_n\textsf{-Collection}$ denote the restriction of the collection scheme to $Π_n$-formulae. In this paper we prove that for $n \geq 1$, if $\mathcal{M}$ is a model of $\mathsf{M}+Π_n\textsf{-Collection}+\mathsf{V=L}$ and $\mathcal{N}$ is a $Σ_{n+1}$-elementary end extension of $\mathcal{M}$ that satisfies $Π_{n-1}\textsf{-Colelction}$ and that contains a new ordinal but no least new ordinal, then $Π_{n+1}\textsf{-Collection}$ holds in $\mathcal{M}$. This result is used to show that for $n \geq 1$, the minimum model of $\mathsf{M}+Π_n\textsf{-Collection}$ has no $Σ_{n+1}$-elementary end extension that satisfies $Π_{n-1}\textsf{-Collection}$, providing a negative answer to the generalisation of a question posed by Kaufmann.

math.LO↗

arXiv2024

Partially-elementary end extensions of countable models of set theory

Let $\mathsf{KP}$ denote Kripke-Platek Set Theory and let $\mathsf{M}$ be the weak set theory obtained from $\mathsf{ZF}$ by removing the collection scheme, restricting separation to $Δ_0$-formulae and adding an axiom asserting that every set is contained in a transitive set ($\mathsf{TCo}$). A result due to Kaufmann shows that every countable model, $\mathcal{M}$, of $\mathsf{KP}+Π_n\textsf{-Collection}$ has a proper $Σ_{n+1}$-elementary end extension. Here we show that there are limits to the amount of the theory of $\mathcal{M}$ that can be transferred to the end extensions that are guaranteed by Kaufmann's Theorem. Using admissible covers and the Barwise Compactness Theorem, we show that if $\mathcal{M}$ is a countable model $\mathsf{KP}+Π_n\textsf{-Collection}+Σ_{n+1}\textsf{-Foundation}$ and $T$ is a recursive theory that holds in $\mathcal{M}$, then there exists a proper $Σ_n$-elementary end extension of $\mathcal{M}$ that satisfies $T$. We use this result to show that the theory $\mathsf{M}+Π_n\textsf{-Collection}+Π_{n+1}\textsf{-Foundation}$ proves $Σ_{n+1}\textsf{-Separation}$.

math.LO↗

arXiv2022

The subset relation and $2$-stratified sentences in set theory and class theory

Hamkins and Kikuchi (2016 and 2017) show that in both set theory and class theory the definable subset ordering of the universe interprets a complete and decidable theory. If $\mathcal{M}$ is a model of set theory, then $\langle M, \subseteq^\mathcal{M} \rangle$ is an atomic unbounded relatively complemented distributive lattice. If $\mathcal{M}$ is model of class theory, then $\langle M, \subseteq^\mathcal{M} \rangle$ is an infinite atomic boolean algebra. We identify the minimal subsystem of $\mathrm{ZF}$, $\mathrm{BAS}$, that ensures that the definable subset relation is an atomic unbounded relatively complemented distributive lattice and classify the atomic unbounded relatively complemented distributive lattices that can be realised as a subset relations of this theory. The fact that the theory of atomic unbounded relatively complemented distributive lattices is complete is used to show that $\mathrm{BAS}$ decides every $2$-stratified sentence of set theory. We also identify the minimal subsystem of $\mathrm{NBG}$, $\mathrm{BAC}$, that ensures that the definable subset relation is an infinite atomic Boolean algebra. We show that there is a complete extension, $\mathrm{IABA}_\mathrm{Ideal}$, of the theory of infinite atomic boolean algebras and an extension $\mathrm{BAC}^+$ of $\mathrm{BAC}$ corresponding to the minimal theory such that if $\mathcal{M}$ is a model of $\mathrm{BAC}^+$, then $\langle M, \mathcal{S}^\mathcal{M}, \subseteq^\mathcal{M} \rangle$ is a model of $\mathrm{IABA}_{\mathrm{Ideal}}$, where $\mathcal{S}^\mathcal{M}$ is a unary predicate that distinguishes sets from classes. This is used to show that $\mathrm{BAC}^+$, a subsystem of $\mathrm{NBG}$, decides every $2$-sentence in the language of class theory that includes a unary predicate distinguishing sets from classes.

math.LO↗

arXiv2022

End extending models of set theory via power admissible covers

Motivated by problems involving end extensions of models of set theory, we develop the rudiments of the power admissible cover construction (over ill-founded models of set theory), an extension of the machinery of admissible covers invented by Barwise as a versatile tool for generalizing model-theoretic results about countable well-founded models of set theory to countable ill-founded ones. Our development of the power admissible machinery allows us to obtain new results concerning powerset-preserving end extensions and rank extensions of countable models of subsystems of $\mathsf{ZFC}$. The canonical extension $\mathsf{KP}^\mathcal{P}$ of Kripke-Platek set theory $\mathsf{KP}$ plays a key role in our work; one of our results refines a theorem of Rathjen by showing that $Σ_1^\mathcal{P}\text{-}\mathsf{Foundation}$ is provable in $\mathsf{KP}^\mathcal{P}$ (without invoking the axiom of choice).

math.LO↗

arXiv2022

Partially-elementary end extensions of countable admissible sets

A result of Kaufmann shows that if $L_α$ is countable, admissible and satisfies $Π_n\textsf{-Collection}$, then $\langle L_α, \in \rangle$ has a proper $Σ_{n+1}$-elementary end extension. This paper investigates to what extent the theory that holds in $\langle L_α, \in \rangle$ can be transferred to the partially-elementary end extensions guaranteed by Kaufmann's result. We show that there are $L_α$ satisfying full separation, powerset and $Π_n\textsf{-Collection}$ that have no proper $Σ_{n+1}$-elementary end extension satisfying either $Π_{n}\textsf{-Collection}$ or $Π_{n+3}\textsf{-Foundation}$. In contrast, we show that if $A$ is a countable admissible set that satisfies $Π_n\textsf{-Collection}$ and $T$ is a recursively enumerable theory that holds in $\langle A, \in \rangle$, then $\langle A, \in \rangle$ has a proper $Σ_n$-elementary end extension that satisfies $T$.

math.LO↗

arXiv2021

Initial self-embeddings of models of set theory

By a classical theorem of Harvey Friedman (1973), every countable nonstandard model $\mathcal{M}$ of a sufficiently strong fragment of ZF has a proper rank-initial self-embedding $j$, i.e., $j$ is a self-embedding of $\mathcal{M}$ such that $j[\mathcal{M}]\subsetneq\mathcal{M}$, and the ordinal rank of each member of $j[\mathcal{M}]$ is less than the ordinal rank of each element of $\mathcal{M}\setminus j[\mathcal{M}]$. Here we investigate the larger family of proper initial-embeddings $j$ of models $\mathcal{M}$ of fragments of set theory, where the image of $j$ is a transitive submodel of $\mathcal{M}$.

math.LO↗

arXiv2019

On the relative strengths of fragments of collection

Let $\mathbf{M}$ be the basic set theory that consists of the axioms of extensionality, emptyset, pair, union, powerset, infinity, transitive containment, $Δ_0$-separation and set foundation. This paper studies the relative strength of set theories obtained by adding fragments of the set-theoretic collection scheme to $\mathbf{M}$. We focus on two common parameterisations of collection: $Π_n$-collection, which is the usual collection scheme restricted to $Π_n$-formulae, and strong $Π_n$-collection, which is equivalent to $Π_n$-collection plus $Σ_{n+1}$-separation. The main result of this paper shows that for all $n \geq 1$, (1) $\mathbf{M}+Π_{n+1}\textrm{-collection}+Σ_{n+2}\textrm{-induction on } ω$ proves the consistency of Zermelo Set Theory plus $Π_{n}$-collection, (2) the theory $\mathbf{M}+Π_{n+1}\textrm{-collection}$ is $Π_{n+3}$-conservative over the theory $\mathbf{M}+\textrm{strong }Π_n \textrm{-collection}$. It is also shown that (2) holds for $n=0$ when the Axiom of Choice is included in the base theory. The final section indicates how the proofs of (1) and (2) can be modified to obtain analogues of these results for theories obtained by adding fragments of collection to a base theory (Kripke-Platek Set Theory with Infinity and $V=L$) that does not include the powerset axiom.

math.LO↗

arXiv2017

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.

math.LO↗

arXiv2016

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.

math.LO↗

arXiv2016

$X$-torsion and universal groups

For a set $X\subseteq \mathbb{N}$, we define the $X$-torsion of a group $G$ to be all elements $g\in G$ with $g^{n}=e$ for some $n\in X$. With $X$ recursively enumerable, we give two independent proofs (group-theoretic, and model-theoretic) that there exists a universal finitely presented $X$-torsion-free group; one which contains all finitely presented $X$-torsion-free groups. We also show that, if $X$ is recursively enumerable, then the set of finite presentations of $X$-torsion-free groups is $Π_{2}^{0}$-complete in Kleene's arithmetic hierarchy.

math.GR↗

arXiv2016

On the strength of a weak variant of the Axiom of Counting

In this paper $\mathrm{NFU}^{-\mathrm{AC}}$ is used to denote Ronald Jensen's modification of Quine's `New Foundations' Set Theory ($\mathrm{NF}$) fortified with a type-level pairing function but without the Axiom of Choice. The axiom $\mathrm{AxCount}_\geq$ is the variant of the Axiom of Counting which asserts that no finite set is smaller than its own set of singletons. This paper shows that $\mathrm{NFU}^{-\mathrm{AC}}+\mathrm{AxCount}_\geq$ proves the consistency of the Simple Theory of Types with Infinity ($\mathrm{TSTI}$). This result implies that $\mathrm{NF}+\mathrm{AxCount}_\geq$ proves that consistency of $\mathrm{TSTI}$, and that $\mathrm{NFU}^{-\mathrm{AC}}+\mathrm{AxCount}_\geq$ proves the consistency of $\mathrm{NFU}^{-\mathrm{AC}}$.

math.LO↗

arXiv2016

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.

math.LO↗

arXiv2014

Decidable fragments of the Simple Theory of Types with Infinity and NF

We identify complete fragments of the Simple Theory of Types with Infinity ($\mathrm{TSTI}$) and Quine's $\mathrm{NF}$ set theory. We show that $\mathrm{TSTI}$ decides every sentence $ϕ$ in the language of type theory that is in one of the following forms: (A) $ϕ= \forall x_1^{r_1} \cdots \forall x_k^{r_k} \exists y_1^{s_1} \cdots \exists y_l^{s_l} θ$ where the superscripts denote the types of the variables, $s_1 > \ldots > s_l$ and $θ$ is quantifier-free, (B) $ϕ= \forall x_1^{r_1} \cdots \forall x_k^{r_k} \exists y_1^{s} \cdots \exists y_l^{s} θ$ where the superscripts denote the types of the variables and $θ$ is quantifier-free. This shows that $\mathrm{NF}$ decides every stratified sentence $ϕ$ in the language of set theory that is in one of the following forms: (A') $ϕ= \forall x_1 \cdots \forall x_k \exists y_1 \cdots \exists y_l θ$ where $θ$ is quantifier-free and $ϕ$ admits a stratification that assigns distinct values to all of the variable $y_1, \ldots, y_l$, (B') $ϕ= \forall x_1 \cdots \forall x_k \exists y_1 \cdots \exists y_l θ$ where $θ$ is quantifier-free and $ϕ$ admits a stratification that assigns the same value to all of the variables $y_1, \ldots, y_l$.

math.LO↗

arXiv2013

Automorphisms of models of set theory and extensions of NFU

In this paper we exploit the structural properties of standard and non-standard models of set theory to produce models of set theory admitting automorphisms that are well-behaved along an initial segment of their ordinals. $\mathrm{NFU}$ is Ronald Jensen's modifcation of Quine's `New Foundations' set theory that allows non-sets into the domain of discourse. The axioms $\mathrm{AxCount}$, $\mathrm{AxCount}_\leq$ and $\mathrm{AxCount}_\geq$ each extend $\mathrm{NFU}$ by placing restrictions on the cardinality of a finite set of singletons relative to the cardinality of its union. Using the results about automorphisms of models of set theory we separate the consistency strengths of these three extensions of $\mathrm{NFU}$. We show that $\mathrm{NFU} + \mathrm{AxCount}$ proves the consistency of $\mathrm{NFU} + \mathrm{AxCount}_\leq$, and $\mathrm{NFU} + \mathrm{AxCount}_\leq$ proves the consistency of $\mathrm{NFU} + \mathrm{AxCount}_\geq$.

math.LO↗