arXiv subjects

John T. Baldwin

Publications and source records attributed to John T. Baldwin.

At least 19 recordsLinked to original sources

arXiv2026

Carnapian Frameworks and Categoricity of Arithmetic via Inferential $ω$-logics

We provided in \cite{BaldwinBrincusI} extensions of first order logic by modified inferential definitions of the classical $ω$-rule in $1$ or $2$ sorts. These logics are categorical in the inferential sense. Arithmetic has a unique countable model in each case, e.g. first order PA is categorical in our first logic. The 2-sorted case interprets $L_{ω_1,ω}$. In this paper, we discuss two philosophical problems raised by Button and Walsh \cite{ButtonWalshbook} concerting the identification of a unique isomorphism class. First, we argue that the doxological challenge (on referential determinacy) gets a clear answer if placed in an appropriate (Carnapian) linguistic framework and is meaningless otherwise. To clarify this approach, we address Button-Walsh's dismissal of concepts-modelism by developing the notion of {\em cognitive modelism}, according to which classical mathematics is a complex process of constructing and developing a distinctive class of concepts. Second, we argue that the inferential $ω$-logics, that are much weaker than second order logic, do not appeal to the arithmetical concepts that the categoricity theorems proved within these logics aim to secure.

math.LO↗

arXiv2026

Categoricity for an inferential $ω$-logic and in $L_{ω_1,ω}$

This paper provides two extensions of first order logic by `$ω$-rules'. In each case we characterize the countable structures whose theory in the logic is categorical (has a unique model). In the one-sorted inferential $ω$-logic, both Robinson's system $Q$ and Peano Arithmetic become categorical. In the two-sorted generalized $ω$-logic we show each complete $L_{ω_1,ω}$ sentence defines the same class of structures as a first-order theory with the appropriate $G-ω$-rule. The results depend on proving that the inferential rules for the logics are categorical, i.e. they uniquely determine certain truth-conditions for the logical connectives and quantifiers.

math.LO↗

arXiv2025

An analogue of U-rank for atomic classes

For a countable, complete, first-order theory $T$, we study $At$, the class of atomic models of $T$. We develop an analogue of $U$-rank and prove two results. On one hand, if some tp(d/a) is not ranked, then there are $2^{\aleph_1}$ non-isomorphic models in $At$ of size $\aleph_1$. On the other hand, if all types have finite rank, then the rank is fully additive and every finite tuple is dominated by an independent set of realizations of pseudo-minimal types.

math.LO↗

arXiv2024

Variations on a Theme of Makowski

We distinguish the axiomatic study of proofs in geometry from study about geometry from general axioms for mathematics. We briefly report on an abuse of that distinction and its unfortunate effect on US high school education. We review a number of 20th century approaches to synthetic geometry. In doing so, we disambiguate (in the Wikipedia sense) the terms: metric, orthogonal, isotropic and hyperbolic. With some of these systems we are able to axiomatize `affine geometry' over the complex field (The argument is trivial from [Wu94] or [Szm78], but not remarked by either of them.). We examine the general question of the connections between axioms for Affine geometries and the stability classification of associated complete first order theories of fields. We conclude with reminiscences of a half-century friendship with Janós.

math.MG↗

arXiv2024

Towards a Finer Classification of Strongly Minimal Sets

Let $M$ be strongly minimal and constructed by a `Hrushovski construction'. If the Hrushovski algebraization function $μ$ is in a certain class ${\mathcal T}$ ($μ$ triples) we show that for independent $I$ with $|I| >1$, ${\rm dcl}^*(I)= \emptyset$ (* means not in ${\rm dcl}$ of a proper subset). This implies the only definable truly $n$-ary function $f$ ($f$ `depends' on each argument), occur when $n=1$. We prove, indicating the dependence on $μ$, for Hrushovski's original construction and including analogous results for the strongly minimal $k$-Steiner systems of Baldwin and Paolini 2021 that the symmetric definable closure, ${\rm sdcl}^*(I) =\emptyset$, and thus the theory does not admit elimination of imaginaries. In particular, such strongly minimal Steiner systems with line-length at least 4 do not interpret a quasigroup, even though they admit a coordinatization if $k = p^n$. The proofs depend on our introduction for appropriate $G \subseteq {\rm aut}(M)$ the notion of a $G$-normal substructure ${\mathcal A}$ of $M$ and of a $G$-decomposition of ${\mathcal A}$. These results lead to a finer classification of strongly minimal structures with flat geometry; according to what sorts of definable functions they admit.

math.LO↗

arXiv2023

Zilber's notion of logically perfect structure: Universal Covers

We sketch recent interactions between model theory and a roughly 150-year old study of analytic functions involving complex analysis, algebraic topology, and number theory, centered in canonicity of universal covers. Towards this goal we discuss in a systematic and unified way several examples indicating the main ideas of the proofs and the necessary changes in method for different situations: exponential covers, modular and Shimura curves, Shimura and abelian varieties, and coherent families of smooth covers.

math.LO↗

arXiv2023

When does $\aleph_1$-categoricity imply $ω$-stability?

For an $\aleph_1$-categorical atomic class, we clarify the space of types over the unique model of size $\aleph_1$. Using these results, we prove that if such a class has a model of size $\beth_1^+$ then it is $ω$-stable.

math.LO↗

arXiv2023

Strongly Minimal Steiner Systems II: Coordinatization and Quasigroups

We note that a strongly minimal Steiner $k$-Steiner system $(M,R)$ from (Baldwin-Paolini 2020) can be `coordinatized' in the sense of (Gantner-Werner 1975) by a quasigroup if $k$ is a prime-power. But for the basic construction this coordinatization is never definable in $(M,R)$. Nevertheless, by refining the construction, if $k$ is a prime power there is a $(2,k)$-variety of quasigroups which is strongly minimal and definably coordinatizes a Steiner $k$-system.

math.LO↗

arXiv2022

Strongly minimal Steiner Systems III: Path graphs and sparse configurations

We introduce a uniform method of proof for the following results. For {\em each} of the following conditions, there are $2^{\aleph_0}$ families of Steiner systems, satisfying that condition: i) Theorem~2.2.4: (extending \cite{Chicoetal}) each Steiner triple system is $\infty$-sparse and has a uniform but not perfect path graph; ii) (Theorem~5.4.2: (extending \cite{CameronWebb}) each Steiner $k$-system (for $k=p^n$) is $2$-transitive and has a uniform path graph (infinite cycles only); iii) Theorem~2.1.5: (extending \cite{Fujiwaramitre}, each is anti-Pasch (anti-mitre); iv) Theorem~3.6 has an explicit quasi-group structure. In each case all members of the family satisfy the same complete strongly minimal theory and it has $\aleph_0$ countable models and one model of each uncountable cardinal.

math.CO↗

arXiv2021

Hanf numbers for Extendibility and related phenomena

This paper contains portions of Baldwin's talk at the Set Theory and Model Theory Conference (Institute for Research in Fundamental Sciences, Tehran, October 2015) and a detailed proof that in a suitable extension of ZFC, there is a complete sentence of $L_{ω_1,ω}$ that has maximal models in cardinals cofinal in the first measurable cardinal and, of course, never again.

math.LO↗

arXiv2021

Maximal models up to the first measurable in ZFC

Theorem: There is a {\em complete sentence} $ϕ$ of $L_{ω_1,ω}$ such that $ϕ$ has maximal models in a set of cardinals $λ$ that is cofinal in the first measurable $μ$ while $ϕ$ has no maximal models in any $χ\geq μ$.

math.LO↗

arXiv2017

Henkin constructions of models with size continuum

We survey the technique of constructing customized models of size continuum in omega steps and illustrate the method by giving new proofs of mostly old results within this rubric. One new theorem, which is joint with Saharon Shelah, is that a pseudominimal theory has an atomic model of size continuum.

math.LO↗

arXiv2015

Constructing many atomic models in $\aleph_1$

We introduce the notion of pseudo-algebraicity to study atomic models of first order theories (equivalently models of a complete sentence of $L_{ω_1,ω}$. Theorem: Let $T$ be any complete first-order theory in a countable language with an atomic model. If the pseudo-minimal types are not dense, then there are $2^{\aleph_1}$ pairwise non-isomorphic atomic models of $T$, each of size $\aleph_1$.

math.LO↗

arXiv2015

The Joint Embedding Property and Maximal Models

We introduce the notion of a `pure` Abstract Elementary Class to block trivial counterexamples. We study classes of models of bipartite graphs and show: Main Theorem (cf. Theorem 3.5.2 and Corollary 3.5.6): If $(λ_i : i \le α<\aleph_1)$ is a strictly increasing sequence of characterizable cardinals (Definition 2.1) whose models satisfy JEP$(<λ_0)$, there is an $L_{ω_1,ω}$ -sentence $ψ$ whose models form a pure AEC and (1) The models of $ψ$ satisfy JEP$(<λ_0)$, while JEP fails for all larger cardinals and AP fails in all infinite cardinals. (2) There exist $2^{λ_i^+}$ non-isomorphic maximal models of $ψ$ in $λ_i^+$, for all $i \le α$, but no maximal models in any other cardinality; and (3) $ψ$ has arbitrarily large models. In particular this shows the Hanf number for JEP and the Hanf number for maximality for pure AEC with Lowenheim number $\aleph_0$ are at least $\beth_{ω_1}$. We show that although AP$(κ)$ for each $κ$ implies the full amalgamation property, JEP$(κ)$ for each κdoes not imply the full joint embedding property. We show the main combinatorial device of this paper cannot be used to extend the main theorem to a complete sentence.

math.LO↗

arXiv1998

On the classifiability of cellular automata

Based on computer simulations Wolfram presented in several papers conjectured classifications of cellular automata into 4 types. He distinguishes the 4 classes of cellular automata by the evolution of the pattern generated by applying a cellular automaton to a finite input. Wolfram's qualitative classification is based on the examination of a large number of simulations. In addition to this classification based on the rate of growth, he conjectured a similar classification according to the eventual pattern. We consider here one formalization of his rate of growth suggestion. After completing our major results (based only on Wolfram's work), we investigated other contributions to the area and we report the relation of some of them to our discoveries.

math.LO↗

arXiv1996

Randomness and semigenericity

Let L contain only the equality symbol and let L^+ be an arbitrary finite symmetric relational language containing L . Suppose probabilities are defined on finite L^+ structures with ''edge probability'' n^{- alpha}. By T^alpha, the almost sure theory of random L^+-structures we mean the collection of L^+-sentences which have limit probability 1. T_alpha denotes the theory of the generic structures for K_alpha, (the collection of finite graphs G with delta_{alpha}(G)=|G|- alpha. | edges of G | hereditarily nonnegative.) THEOREM: T_alpha, the almost sure theory of random L^+-structures is the same as the theory T_alpha of the K_alpha-generic model. This theory is complete, stable, and nearly model complete. Moreover, it has the finite model property and has only infinite models so is not finitely axiomatizable.

math.LO↗

arXiv1996

DOP and FCP in generic structures

Spencer and Shelah [ShSp:304] constructed for each irrational alpha between 0 and 1 the theory T^alpha as the almost sure theory of random graphs with edge probability n^{- alpha}. In [BlSh:528] we proved that this was the same theory as the theory T_alpha built by constructing a generic model in Baldwin and Shi. In this paper we explore some of the more subtle model theoretic properties of this theory. We show that T^alpha has the dimensional order property and does not have the finite cover property.

math.LO↗

arXiv1995

Abstract classes with few models have `homogeneous-universal' models

This paper is concerned with a class K of models and an abstract notion of submodel <=. Experience in first order model theory has shown the desirability of finding a `monster model' to serve as a universal domain for K. In the original constructions of Jonsson and Fraisse, K was a universal class and ordinary substructure played the role of <=. Working with a cardinal lambda satisfying lambda^{< lambda}= lambda guarantees appropriate downward Lowenheim-Skolem theorems; the existence and uniqueness of a homogeneous-universal model appears to depend centrally on the amalgamation property. We make this apparent dependence more precise in this paper. The major innovation of this paper is the introduction of weaker notion to replace the natural notion of (K, <=)-homogeneous-universal model. Modulo a weak extension of ZFC (provable if V=L), we show that a class K obeying certain minimal restrictions satisfies a fundamental dichotomy: For arbitrarily large lambda, either K has the maximal number of models in power lambda or K has a unique chain homogenous-universal model of power lambda. We show that in a class with amalgamation this dichotomy holds for the notion of K-homogeneous-universal model in the more normal sense.

math.LO↗