Below, you may find a collection of online preprints, separated into five rough categories.

**Proof Theory.**This includes work on dilators and Pi12-logic, finitary and infinitary proof systems, complexity and cut elimination, and logics of proofs.

**Recursion Theory.**

**Set Theory.**

**Model Theory.**This includes work involving completeness theorems, Löwenheim-Skolem theorems, Hanf numbers, and related notions.

**Other work.**

- Ackermann and Goodstein Go Functorial (with Anton Freund, Michael Rathjen, and Andreas Weiermann).

**Pacific J. Math.**, in press.

__The Number of Axioms__(with Matthias Baaz and Jan Bydžovský).

*Submitted for publication.*

__Determinate Logic and the Axiom of Choice__.

**Ann. Pure Appl. Logic**171 (2020).

__Unsound Inferences Make Proofs Shorter__(with Matthias Baaz).

**J. Symbolic Logic**84 (2019), pp. 102-122*.*

__Verification Logic__(with David Fernández-Duque).

**J. Log. Comput.**27 (2017), pp. 2451-2469.

- A proof of Grilliot's Theorem (with Ferre Vanden Kerchove).

*Submitted for publication.*

__Effective Cardinals and Determinacy in Third-Order Arithmetic.__

*Submitted for publication.*

- Feedback Hyperjump (with Robert Lubarsky).

**J. Log. Comput.**31 (2021), pp. 20-39.

- A Characterization of Σ11-Reflecting Ordinals.

**Ann. Pure Appl. Logic,**

- The Order of Reflection.

**J. Symbolic Logic**, in press.

__Boundedness Theorems for Flowers and Sharps__(with Anton Freund, Michael Rathjen, and Andreas Weiermann).

**Proc. Amer. Math. Soc.**, in press.

__Countable Ordinals in Indiscernibility Spectra.__

*Submitted for publication.*

- Gδσ Games and Induction on Reals (with Philip Welch).

**J. Symbolic Logic**, in press.

- Projective Games on the Reals (with Sandra Müller).

**Notre Dame J. Form. Log.**64 (2020) pp. 573–589.

- Provably Δ1 Games (with Douglas Blue).

**J. Symbolic Logic**85 (2020), pp. 1124–1146.

- Fσ Games and Reflection in L(R).

**J. Symbolic Logic**85 (2020), pp. 1102–1123.

- Long Borel Games.

**Israel J. Math.**, in press.

- Shortening Clopen Games.

**J. Symbolic Logic**, in press.

- Determined Admissible Sets.

**Proc. Amer. Math. Soc.**148 (2020), pp. 2217-2231.

__The Consistency Strength of Long Projective Determinacy__(with Sandra Müller).

**J. Symbolic Logic**85 (2020), pp. 338-366.

__Long Games and__σ__-Projective Sets__(with Sandra Müller and Philipp Schlicht).

**Ann. Pure Appl. Logic,**172 (2021).

- σ
__-Projective Determinacy__.

*Submitted for publication.*

- The Löwenheim-Skolem Theorem for Gödel Logic.

*Submitted for publication.*

__A Topological Completeness Theorem for Transfinite Provability Logic__.

*Submitted for publication.*

__Strong Completeness of Provability Logic for Ordinal Spaces__(with David Fernández-Duque).

**J. Symbolic Logic**82 (2017), pp. 608-628.

__On the Maximization of Menu-Dependent Interval Orders__(with Levent Ülkü).

**S. Choice Welf. 48 (**2017), pp. 357-366.

__Ten Problems in Gödel Logic__(with Matthias Baaz).

**Soft Comput.**21 (2017), pp. 149-152.

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