•  8
    Corrigendum to Reducing ω-model reflection to iterated syntactic reflection
    with Fedor Pakhomov
    Journal of Mathematical Logic 23 (3). 2023.
    We fix a gap in a proof in our paper Reducing ω-model reflection to iterated syntactic reflection.
  •  106
    Evitable iterates of the consistency operator
    Computability 12 (1): 59--69. 2023.
    Why are natural theories pre-well-ordered by consistency strength? In previous work, an approach to this question was proposed. This approach was inspired by Martin's Conjecture, one of the most prominent conjectures in recursion theory. Fixing a reasonable subsystem $T$ of arithmetic, the goal was to classify the recursive functions that are monotone with respect to the Lindenbaum algebra of $T$. According to an optimistic conjecture, roughly, every such function must be equivalent to an iterat…Read more
  •  14
    Characterizations of ordinal analysis
    Annals of Pure and Applied Logic 174 (4): 103230. 2023.
    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 equival…Read more
  •  4
    A note on the consistency operator
    Proceedings of the American Mathematical Society 148 (6): 2645--2654. 2020.
    It is a well known empirical observation that natural axiomatic theories are pre-well-ordered by consistency strength. For any natural theory $T$, the next strongest natural theory is $T+\mathsf{Con}_T$. We formulate and prove a statement to the effect that the consistency operator is the weakest natural way to uniformly extend axiomatic theories.
  •  4
    Incompleteness and jump hierarchies
    with Patrick Lutz
    Proceedings of the American Mathematical Society 148 (11): 4997--5006. 2020.
    This paper is an investigation of the relationship between G\"odel's second incompleteness theorem and the well-foundedness of jump hierarchies. It follows from a classic theorem of Spector's that the relation $\{(A,B) \in \mathbb{R}^2 : \mathcal{O}^A \leq_H B\}$ is well-founded. We provide an alternative proof of this fact that uses G\"odel's second incompleteness theorem instead of the theory of admissible ordinals. We then derive a semantic version of the second incompleteness theorem, origin…Read more
  •  40
    An incompleteness theorem via ordinal analysis
    Journal of Symbolic Logic 1-17. forthcoming.
    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 an…Read more
  •  18
    Reducing omega-model reflection to iterated syntactic reflection
    with Fedor Pakhomov
    Journal of Mathematical Logic 23 (2). 2021.
    Journal of Mathematical Logic, Volume 23, Issue 02, August 2023. 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 …Read more
  •  16
    Reflection ranks and ordinal analysis
    with Fedor Pakhomov
    Journal of Symbolic Logic 86 (4): 1350-1384. 2021.
    It is well-known that natural axiomatic theories are well-ordered by consistency strength. However, it is possible to construct descending chains of artificial theories with respect to consistency strength. We provide an explanation of this well-orderedness phenomenon by studying a coarsening of the consistency strength order, namely, the$\Pi ^1_1$reflection strength order. We prove that there are no descending sequences of$\Pi ^1_1$sound extensions of$\mathsf {ACA}_0$in this ordering. According…Read more
  •  24
    On the inevitability of the consistency operator
    with Antonio Montalbán
    Journal of Symbolic Logic 84 (1): 205-225. 2019.
    We examine recursive monotonic functions on the Lindenbaum algebra of EA. We prove that no such function sends every consistent φ to a sentence with deductive strength strictly between φ and (φ∧Con(φ)). We generalize this result to iterates of consistency into the effective transfinite. We then prove that for any recursive monotonic function f, if there is an iterate of Con that bounds f everywhere, then f must be somewhere equal to an iterate of Con