Workshop: Variations on Metamathematics

In 1960, Feferman first called attention to the notion of intensionality, or indeterminacy, in metamathematics. This notion gave rise to a variety of mathematical and philosophical questions related to coding, provability, truth, and other metamathematical concepts.

The Variations on Metamathematics workshop aims to discuss some of these questions. It is organised within Léon Probst’s SNSF Doc.CH Grant (#227030).

Date: 3 – 4 July 2025

Venue: 

SI-006, Black Building
USI West Campus
Via Buffi 13
Lugano, Switzerland

Speakers:

Balthasar Grabmayr (University of Tübingen)
Volker Halbach (University of Oxford)
Carlo Nicolai (King’s College London)
Lavinia Picollo (National University of Singapore)
Léon Probst (University of Lugano)
Albert Visser (University of Utrecht)
Daniel Waxman (National University of Singapore)

Programme:

Thursday 3rd:

10.30 - 10.45 Welcome
10:45 – 12:00 V. Halbach
Lunch Break
14.00 - 15.15 L. Picollo and D. Waxman
15:15 - 16:30 B. Grabmayr
16.30 - 16.45 Coffee Break
16.45 - 18.00 L. Probst

Friday 4th:

09:00 - 10:15 A. Visser
10:15 - 10:30 Coffee break
10:30 - 11:45 C. Nicolai
11:45 - 12:15 Final discussion

Abstracts:

Balthasar Gramayr: Computation and the Structure of Arithmetic
Abstract: It is a truism of mathematics that differences between isomorphic number systems are irrelevant to arithmetic. This truism is deeply rooted in the modern axiomatic method and underlies most strands of arithmetical structuralism, the view that arithmetic is about some abstract number structure. In this talk, I challenge this truism by showing that isomorphic systems can differ with regard to important computational features of numbers. This confronts arithmetical structuralists with a dilemma. On the one hand, many computability-theoretic properties are only satisfied by particular number systems, and are hence disqualified as irrelevant by structuralist accounts. On the other hand, these properties turn out to be highly relevant to arithmetical practice. After presenting this dilemma, I will discuss the consequences for structuralism as a view about arithmetic.

Volker Halbach: Logical consequence and reflection
Abstract: I try to connect various topics in metamathematics on the background of my axiomatic theory of Classical and Determinate Truth. In particular, I relate proof-theoretic reflection principles to proofs of soundness and completeness of logic. The primitive axiomatized truth predicate will prove to be central to a proper understanding of all these phenomena and their interactions.

Carlo Nicolai: Disentangled syntax, type-free
Abstract: Theories of truth with disentangled syntax provide a versatile method to formulate a theory of truth for a variety of languages, including the ones that are not sufficiently expressive to formalize syntactic notions and operations. So far, disentanglement of syntax has primarily concerned typed, Tarskian theories of truth. In the talk, I discuss the prospect of type-free theories of disentangled syntax and prove some preliminary results.

Lavinia Picollo and Daniel Waxman: Can we follow the omega rule?
Abstract: A much-discussed problem in the philosophy of mathematics concerns how our mathematical practice, broadly construed, could possibly give rise to determinate truth values for all sentences in the language of arithmetic. One proposal, which dates originally to Rudolf Carnap and has recently been developed by Jared Waren, is that we can and do in fact follow the omega rule -- an infinitary rule that yields a complete theory of arithmetic. This talk will critically examine the case that finite beings like us can follow infinitary rules like the omega rule. 

Léon Probst: On the choice of numbering
Abstract: Numberings have been identified as one of the sources of indeterminacy in metamathematics. However, it is widely believed that our metamathematical theorems hold for all reasonable numberings. Even if we have strong intuitions about whether a particular numbering is reasonable, it is far from obvious to explain why - let alone provide a formal definition. In this talk, I focus on the choice of numbering and how to deal with the question of reasonable numbering. I discuss the two standard approaches (resemblance and meaning conditions) used in the literature to address indeterminacy in metamathematics and relate them to numberings.

Albert Visser: Sequentiality, Markov coding, Gödel Numbering
Abstract: This talk is a sequel to my talk in Oxford of September 2023.
Gödel’s original proof of the sequentiality of arithmetic is  still one of the best ways to build sequences over arithmetic with addition and multiplication. Emil Jeřábek shows that we already have the beta-function with its good properties in an arithmetical theory that is even weaker than the usual base theory PA^-.
There are however other ways of building sequences. Two strategies are based on translating (binary) strings into arithmetic. The first one is due to Raymond Smullyan and the second one is due to Andrej Markov jr. Both coding strategies are strikingly different.
In this talk we discuss the Markov strategy.  The basic idea is that the special linear monoid of N is isomorphic to the binary strings. We show that the Markov strategy can be made to work in PA^- plus the Euclidean Division Axiom. We consider two recursive models of PA^- and reflect on what the Markov strings in these models look like. The results suggest that there is a mystery extension of PA^- that is better for our purposes than our extension with Euclidean Division, but currently I have no good proposal on what it might be.
Tempore volente, we end with a discussion of Gödel numberings in PA^-.

Organisers:

Léon Probst with his SNSF Doc.CH “Intensionality in Metamathematics” (#227030)
An event of the Institute of Philosophy (ISFI)

For more info: [email protected]