James Walsh’s research while affiliated with Cornell University and other places

Ranking theories according to their strength is a recurring motif in mathematical logic. We introduce a new ranking of arbitrary (not necessarily recursively axiomatized) theories in terms of the encoding power of their $\beta$-models: $T\prec_\beta U$ if every $\beta$-model of U contains a countable coded $\beta$-model of T. The restriction of $\prec_\beta$ to theories with $\beta$-models is well-founded. We establish fundamental properties of the attendant ranking. First, though there are continuum-many theories, every theory has countable $\prec_\beta$-rank. Second, the $\prec_\beta$-ranks of $\mathcal{L}_\in$ theories are cofinal in $\omega_1$. Third, assuming V=L, the $\prec_\beta$-ranks of $\mathcal{L}_2$ theories are cofinal in $\omega_1$. Finally, $\delta^1_2$ is the supremum of the $\prec_\beta$-ranks of finitely axiomatized theories.

We fix a gap in a proof in our paper Reducing[Formula: see text]-model reflection to iterated syntactic reflection.

Ordinal analysis is a research program wherein recursive ordinals are assigned to axiomatic theories. According to conventional wisdom, ordinal analysis measures the strength of theories. Yet what is the attendant notion of strength? In this paper we present abstract characterizations of ordinal analysis that address this question. First, we characterize ordinal analysis as a partition of $\Sigma^1_1$-definable and $\Pi^1_1$-sound theories, namely, the partition whereby two theories are equivalent if they have the same proof-theoretic ordinal. We show that no equivalence relation $\equiv$ is finer than the ordinal analysis partition if both: (1) $T\equiv U$ whenever T and U prove the same $\Pi^1_1$ sentences; (2) $T\equiv T+U$ for every set U of true $\Sigma^1_1$ sentences. In fact, no such equivalence relation makes a single distinction that the ordinal analysis partition does not make. Second, we characterize ordinal analysis as an ordering on arithmetically-definable and $\Pi^1_1$-sound theories, namely, the ordering wherein $T< U$ if the proof-theoretic ordinal of T is less than the proof-theoretic ordinal of U. The standard ways of measuring the strength of theories are consistency strength and inclusion of $\Pi^0_1$ theorems. We introduce analogues of these notions -- $\Pi^1_1$-reflection strength and inclusion of $\Pi^1_1$ theorems -- in the presence of an oracle for $\Sigma^1_1$ truths, and prove that they coincide with the ordering induced by ordinal analysis.

We present an analogue of Gödel’s second incompleteness theorem for systems of second-order arithmetic. Whereas Gödel showed that sufficiently strong theories that are $\Pi ^0_1$ -sound and $\Sigma ^0_1$ -definable do not prove their own $\Pi ^0_1$ -soundness, we prove that sufficiently strong theories that are $\Pi ^1_1$ -sound and $\Sigma ^1_1$ -definable do not prove their own $\Pi ^1_1$ -soundness. Our proof does not involve the construction of a self-referential sentence but rather relies on ordinal analysis.

Let's fix a reasonable subsystem T of arithmetic; why are natural extensions of T pre-well-ordered by consistency strength? In previous work, an approach to this question was proposed. The goal of this work was to classify the recursive functions that are monotone with respect to the Lindenabum algebra of T. According to an optimistic conjecture, roughly, every such function must be equivalent to an iterate $\mathsf{Con}_T^\alpha$ of the consistency operator in the limit. In previous work the author established the first case of this optimistic conjecture; roughly, every recursive monotone function is either as weak as the identity operator in the limit or as strong as $\mathsf{Con}_T$ in the limit. Yet in this note we prove that this optimistic conjecture fails already at the next step; there are recursive monotone functions that are neither as weak as $\mathsf{Con}_T$ in the limit nor as strong as $\mathsf{Con}_T^2$ in the limit. In fact, for every $\alpha$, we produce a function that is cofinally as strong as $\mathsf{Con}^\alpha_T$ yet cofinally as weak as $\mathsf{Con}_T$.

It is well-known that natural axiomatic theories are pre-well-ordered by logical strength, according to various characterizations of logical strength such as consistency strength and inclusion of $\Pi^0_1$ theorems. Though these notions of logical strength coincide for natural theories, they are not generally equivalent. We study analogues of these notions -- such as $\Pi^1_1$-reflection strength and inclusion of $\Pi^1_1$ theorems -- in the presence of an oracle for $\Sigma^1_1$ truths. In this context these notions coincide; moreover, we get genuine pre-well-orderings of axiomatic theories and may drop the non-mathematical quantification over "natural" theories.

Ordinal analysis induces a partition of $\Sigma^1_1$-definable and $\Pi^1_1$-sound theories whereby two theories are equivalent if they have the same proof-theoretic ordinal. We show that no equivalence relation $\equiv$ is finer than the ordinal analysis partition if both: (1) $T\equiv U$ whenever T and U prove the same $\Pi^1_1$ sentences; (2) $T\equiv T+U$ for every set U of true $\Sigma^1_1$ sentences. In fact, no such equivalence relation makes a single distinction that the ordinal analysis partition does not make.

jats:p> In mathematical logic there are two seemingly distinct kinds of principles called “reflection principles.” Semantic reflection principles assert that if a formula holds in the whole universe, then it holds in a set-sized model. Syntactic reflection principles assert that every provable sentence from some complexity class is true. In this paper, we study connections between these two kinds of reflection principles in the setting of second-order arithmetic. We prove that, for a large swathe of theories, [Formula: see text]-model reflection is equivalent to the claim that arbitrary iterations of uniform [Formula: see text] reflection along countable well-orderings are [Formula: see text]-sound. This result yields uniform ordinal analyzes of theories with strength between [Formula: see text] and [Formula: see text]. The main technical novelty of our analysis is the introduction of the notion of the proof-theoretic dilator of a theory [Formula: see text], which is the operator on countable ordinals that maps the order-type of [Formula: see text] to the proof-theoretic ordinal of [Formula: see text]. We obtain precise results about the growth of proof-theoretic dilators as a function of provable [Formula: see text]-model reflection. This approach enables us to simultaneously obtain not only [Formula: see text], [Formula: see text] and [Formula: see text] ordinals but also reverse-mathematical theorems for well-ordering principles. </jats:p

We present an analogue of G\"{o}del's second incompleteness theorem for systems of second-order arithmetic. Whereas G\"{o}del showed that sufficiently strong theories that are $\Pi^0_1$-sound and $\Sigma^0_1$-definable do not prove their own $\Pi^0_1$-soundness, we prove that sufficiently strong theories that are $\Pi^1_1$-sound and $\Sigma^1_1$-definable do not prove their own $\Pi^1_1$-soundness. Our proof does not involve the construction of a self-referential sentence but rather relies on ordinal analysis.

By a previous result of the authors there are no infinite sequences of $\Pi^1_1$-sound extensions of $\mathsf{ACA}_0$ such that each of them proves $\Pi^1_1$-reflection of the next one. This engenders a well-founded "reflection ranking" of $\Pi^1_1$-sound extensions of $\mathsf{ACA}_0$. In previous work the authors proved that for any $\Pi^1_1$-sound theory T extending $\mathsf{ACA}^+_0$, the reflection rank of T equals the proof-theoretic ordinal of T. This provides an alternative characterization of the notion of "proof-theoretic ordinal," which is among the central concepts in proof theory. The authors proved this equivalence with techniques from the study of iterated reflection principles. In this note we provide a new proof that instead makes use of traditional proof-theoretic techniques, namely, cut-elimination for infinitary derivations.