Meetings

Potentialism has become a popular conception, also due to the work of Parsons and Linnebo. More recently connections to predicativism have been discussed in the literature. In the talk I discuss Feferman's theory of unfolding from a potentialist perspective. The modality involved will have a constructive flavor. I will focus on the intermediate unfolding and propose a reading as a variation offirst-orderism.

In this talk, I will examine the extent to which the number of self-interpretations of a theory is suitable as a measure of semantic determinacy of theories. The first part will deal with theories that have unique self-interpretations. Examples will be discussed and the connection to other concepts of semantic determinacy will be considered. The second part will deal with theories that have a finite number of self-interpretations and the desirable properties that follow from this for concepts of theoretical equivalence.

Recent work in algebra-valued models of set theory reveals a striking form of foundational indeterminacy: there is no unique logic determined by the mathematical role of set theory itself. Within a single uniform construction—the cumulative hierarchy parameterized by different algebraic truth-value structures—we can build classical, intuitionistic, paraconsistent, fuzzy, and quantum models of ZF-like set theory. Remarkably, many of these models validate axiom systems classically equivalent to ZF and contain faithful copies of the ordinary cumulative hierarchy, allowing large portions of standard mathematics to be recovered internally. 

This yields a Benacerraf-style dilemma for logical monism. If two set theories, one classical and one non-classical, validate the same axioms, interpret the same language, and support the same mathematical practice, then no purely mathematical criterion selects one underlying logic as “the” correct logic of mathematics. 

Foundational correctness is therefore not fixed by classicality, but by a deeper notion of convergence: distinct logics can generate set-theoretic universes that converge in mathematical expressiveness. I argue that this convergence establishes a precise and technically grounded form of logical pluralism.

Accounts of abstract reference that derive from Frege, as well as appealing to abstraction principles, are concerned with whether a term functions as a singular term in suitable true sentences and whether it is associated with a statement that gives identity conditions for the corresponding object.

The aim of this paper is to clarify those three principles of abstract reference and the relations between them. I argue, moreover, that reflection on them brings to light a discriminating potential that the Fregean views have to offer – between cases where reference is successful and where it is not. The example that I use, of apparent reference that is, in fact, not successful, is that of fictional names.

In this talk, we will take an eagles eyed vie of the biinterpretability proofs between the theories of KFμ (also known as Kripke-Feferman-Burgess), the first order theory of inductive definitions of arithmetic ID1 (mostly motivated from proof theory) and the set theory KP that has seen applications in higher recursion theory via the study of admissible set theory. In particular, the version of KP we will use will be what we will dub as KPs (KP small) which constitutes KP + infinity + V=Lω_1^CK  (i.e. a theory of the Hyperelementary sets).

This connection enables us to freely use theorems of admissible set theory like Barwise Compactness to construct non-standard models of KFμ and ID_1 whose first order part is standard.

During the talk we present the outcome of a recent joint work with Piotr Gruza devoted to various criteria of scheme definiteness. The project can be seen as the next natural step in the investigation of the general notion of internal categoricity. The latter concept was studied by Vaananen ("Tracing Internal Categoricity", "An Extension of a Theorem of Zermelo") in the context of PA and ZF and applied to the problem of semantical determinacy of the statements of arithmetic and set theory by Button and Walsh (Philosophy and Model Theory). Later on, in "Categoricity-like Properties in the First-Order Realm" (by Enayat and the author), it was argued that internal categoricity is best seen as a property of schemes, and some preliminary results about the behaviour of this notion were obtained.

We start the talk by explaining the philosophical motivations for this kind of research (relying mainly on Philosophy and Model Theory). Then we introduce a general template from which various notions of definiteness (such as strong internal categoricity, internal categoricity, intolerance) can be obtained. A new feature of our definition is that we introduce a parameter that allows us to measure the amount of resources needed to derive the respective definiteness claim. We show that our definition behaves well with respect to a natural retract relation between schemes (which is obtained naturally from a new notion of scheme interpretability). We separate various definiteness notions using fairly natural sequential theories. Last but not least, we apply the new definitions in the context of two canonical foundational theories: PA and ZF.

Petr Vopěnka (1935--2015) was an influential mathematician and deep and original thinker. In the 1970s, he and Petr Hájek developed a system  of  non-Cantorial set theory, which was first known the theory of semi-sets, and later became Alternative Set Theory (AST). The primary new  concept on which  the axioms of AST are based is what Vopěnka called natural infinity. In my talk I  will outline some of Vopěnka's thoughts on foundations of mathematics and the  role of set theory in it, followed by discussion of the axioms of AST.

The concept of a punctual (or fully primitive recursive) structure has received a lot of attention in recent years. It constitutes a refinement of the notion of a computable structure, requiring the domain to be the set of all natural numbers and all functions and relations from the signature to be primitive recursive.

A key difference between computable and punctual structures is that the structure (N,S) of natural numbers with successor, while computably categorical, is not punctually categorical and has many punctual isomorphic copies with wildly varying properties. I am going to discuss various results obtained together with Nikolay Bazhenov, Ivan Georgiev, Dariusz Kalociński, Luca San Mauro and Stefan Vatev regarding properties of such copies. The main focus is going to be on characterising classes of functions whose isomorphic images are primitive recursive in various copies of (N,S).

In the paper "Classical Determinate Truth I" Fujimoto and Halbach had introduced a theory CD in which the semantic notions of truth and determinateness are axiomatised as primitive predicates. The truth predicate of that theory is fully classically compositional for the sentences of the whole language and it satisfies Tarski biconditionals for all the determinate sentences. The authors had asked whether that theory can be consistently extended with an axiom saying that for any sentence it is determinate whether it is determinate.

In our talk, we present a recent result obtained by Castaldo, Głowacki, and the author, showing that this extension of CD is in fact inconsistent. We discuss how this result forms a general impossibility result for a broad class of theories of determinateness or groundedness.  

In this talk, we present the construction of non-classical models for ZFC. This is achieved by generalizing the Boolean-valued model construction to algebras that are not necessarily tied to classical logic. We will trace the origins of this approach—from the early constructions validating only fragments of ZFC to more recent results establishing the validity of all axioms of ZFC within a non-classical framework. These models will then be used to extend independence results to non-classical set theories. We will conclude with a philosophical reflection on pluralism in the foundations of set theory.

Around 2002 Leonid Levin introduced his Independence Postulate (IP), which says (roughly speaking) that the mutual algorithmic information, between any pair of a mathematically defined sequence and a physical sequence, should be small. IP can be seen as a stronger, finitary version of the Church Thesis. I will give a gentle introduction to algorithmic information theory, hopefully sufficient to understand IP. Then I will try to reconstruct Levin's argument that, assuming IP, working mathematicians cannot hope to obtain (a large prefix of) a consistent completion of PA, even using non-mechanistic means such as informal arguments, adding new axioms and so on. The talk will be based on https://arxiv.org/pdf/cs/0203029

It is a standard view in the Philosophy of Mathematics that arithmetic is about a unique subject matter, the natural numbers, and that arithmetical truth is (to some degree) determinate. The uniqueness of the arithmetical subject matter is usually understood in terms of categoricity, i.e., that arithmetic has one intended structure (up to isomorphism). Determinacy of truth is understood as the claim that all arithmetical statements are either true or false (in a sense, all arithmetical questions have, in principle, a definite answer). Unfortunately, standard ways employed to prove and argue for the uniqueness and determinacy of arithmetic have turned out to be philosophically problematic. From these issues, a 'challenge' of explaining the uniqueness and determinacy of arithmetic arises. 

Mathematical Internalism promises to overcome the major issues resulting from the standard approach to uniqueness and determinacy. In particular, Internalism aims to explain the determinacy of arithmetical truth by arguing for the intolerance of arithmetic: All arithmetical structures must agree (to some extent) on what is true. Recently, the internalist explanation of determinacy has been criticised. If successful, this critique puts internalism in danger of not being able to meet the challenge of explaining arithmetical determinacy. This talk aims to discuss the internalist argument for the determinacy of arithmetic, analyse its critique and propose an argument for determinacy that avoids the previous critique.

This talk is about Tarski-style satisfaction/truth theories, and their model-theoretic counterparts, i.e., the so-called satisfaction/truth classes. I will focus on a property that has come to be known as "disjunctive correctness", which is the formal counterpart of the "obvious" statement "a disjunction D of arbitrary

finite length is true iff one of the disjuncts of D is true".  We have known since the joint work of Henryk Kotlarski, Stanisłav Krajewski, and Alistair Lachlan (1981) that the compositional theory of truth over PA (Peano Arithmetic) is conservative over PA, i.e., any arithmetical statement provable in the former theory is provable in the latter theory. This result has captured the imagination of philosophical logicians, especially in connection with the debate concerning the deflationist conception of truth. However, up to a few years ago it was unknown whether the result of augmenting the compositional theory of truth over PA with the statement "the truth predicate is disjunctively correct" remains conservative over PA. 

The joint work of the author with Fedor Pakhomov (2019), which was substantially refined by the joint work of Cezary Cieśliński, Mateusz Łełyk, and Bartosz Wcisło (2023) has made it clear that the answer is

in the negative. In the first half of the talk I will review the aforementioned developments, and in the second half I will discuss the main results of my recent paper "Satisfaction classes with approximate disjunctive

correctness" (freely available via: https://link.springer.com/article/10.1007/s00153-018-0657-9) that show

the conservativity of the compositional theory of truth over PA with arbitrarily high approximations of disjunctive correctness.

The talk is devoted to the exposition of recent results revolving around the following vague question

Are salient foundational theories such as Peano Arithmetic (PA) or Zermelo-Frankel set theory (ZF) distinguishable from other theories by virtue of being "more categorical"?

Obviously any theory that can establish the basic truths about addition and multiplication of natural numbers is far from being categorical, in the traditional sense of this term. However, the recent work of Albert Visser, Ali Enayat (tightness and solidity) and Jouko V\"a\"an\"anen (internal categoricity) delivers a bunch of interesting candidates for categoricity-flavoured criteria which can be meaningfully applied to theories of foundations of mathematics. 

In the talk we present a roadmap for these various categoricity-like notions and introduce recent results about them obtained in collaboration with Ali Enayat, Piotr Gruza and Leszek Kołodziejczyk. We argue that the categoricity-like notions are best seen as properties of schemes as opposed to first-order theories. The talk is based on the recent joint paper with Ali Enayat "Categoricity-like properties in the first-order realm".

Our objective is to present a construction of a truth class for the language of first-order arithmetic. In the construction, the notion of a proof approximation will be crucially used. The technique of approximations will be developed as a part of proof theory.

A popular position in philosophy of mathematics is structuralism which states that mathematics is a study of structures and that only structural properties of mathematical objects are relevant. This is in line with a common mathematical practice to treat isomorphic objects as essentially being the same object. Such an approach unfortunately neglects computability theory where isomorphic objects can have very different properties.

This was observed (maybe not explicitly) by Stewart Shapiro who analysed the concept of a notation and showed how a change in a notation we use to represent natural numbers results in different classes of functions being computable. A more general analysis of this phenomenon is presented by a branch of mathematics called computable structure theory which asks a question how different computable isomorphic copies of a structure differ from each other.

In this talk I would like to focus on a certain subarea of CST called punctual structure theory which considers structures with an even stricter condition on computability. We ask that the domain is the entire set N and that all functions and relations in the signature are primitive recursive. This corresponds to algorithms in which unbounded search is ruled out. I would like to discuss philosophical justification and the advantages of such a perspective.

In a less philosophical part of the presentation I would like to consider the question which type of structures have a punctual presentation. This is a very broad topic with lots of interesting results. I intend to discuss at least some of them.

It is well-known that Zermelo-Fraenkel set theory ZFC does not decide many natural questions arising on its grounds. For instance, it fails to answer whether Continuum, the cardinality of real numbers, is the least uncountable cardinal (thus, it cannot decide whether Continuum Hypothesis, CH, holds). On the other hand, by Goedel's theorem it also does not decide whether ZFC is consistent. However, intuitively it seems that the two cases are really different. While CH seems to have a truly undetermined status, it seems that the consistency statement is somehow already decided. One way to distinguish between the two cases is to argue that accepting consistency of ZFC is already implicit in our acceptance of ZFC. The so-called implicit commitment thesis argues precisely in favour of the opinion that in accepting strong theories we are rationally required to accept some additional statements, like consistency or reflection. In our talk, we will discuss a possibility that there is another feature distinguishing these two cases. As opposed to the consistency statements, the status of CH can change between well-founded models of set theory. However, a large class of statements remains fixed between such models and the class of the fixed facts becomes even larger if we accept certain further extensions of the axioms of set theory. We will discuss these so-called absoluteness phenomena and analyse whether they might be relevant for the debate on the status of second-order logic.