arXiv subjects

Jouko Väänänen

Publications and source records attributed to Jouko Väänänen.

At least 37 records · Page 2Linked to original sources

arXiv2023

Philosophical Uses of Categoricity Arguments

Mathematicians and philosophers have appealed to categoricity arguments in a surprisingly varied range of contexts. One familiar example calls on second-order categoricity in an attempt to show that the Continuum Hypothesis, despite its formal independence, has a determinate truth value, but this does not exhaust the uses of categoricity even in set theory, not to mention its appearance in various roles in discussions of arithmetic. Here we compare and contrast a sampling of these deployments to get a sense of when these arguments tend to succeed and when they tend to fail. Our story begins with two historical landmarks, Dedekind and Zermelo, on arithmetic and set theory, respectively, and ends with two leading contemporary writers, Charles Parsons and the co-authors Tim Button and Sean Walsh, again on arithmetic and set theory, respectively. In between, we pause over the well-known contribution of Georg Kreisel. In each case we ask: what does the author set out to accomplish, philosophically?; what do they actually do (or what can be done), mathematically?; and does what is done (or can be done) accomplish what they set out to do? We find this focus on context illuminating: these authors have qualitatively different philosophical goals, and what works for one might not work for another.

math.LO↗

arXiv2023

Sort logic and foundations of mathematics

I have argued elsewhere that second order logic provides a foundation for mathematics much in the same way as set theory does, despite the fact that the former is second order and the latter first order, but second order logic is marred by reliance on ad hoc {\em large domain assumptions}. In this paper I argue that sort logic, a powerful extension of second order logic, provides a foundation for mathematics without any ad hoc large domain assumptions. The large domain assumptions are replaced by ZFC-like axioms. Despite this resemblance to set theory sort logic retains the structuralist approach to mathematics characteristic of second order logic. As a model-theoretic logic sort logic is the strongest logic. In fact, every model class definable in set theory is the class of models of a sentence of sort logic. Because of its strength sort logic can be used to formulate particularly strong reflection principles in set theory.

math.LO↗

arXiv2022

An atom's worth of anonymity

Anonymity has gained notoriety in modern times as data about our actions and choices accumulates in the internet partly unbeknownst to us and partly by our own choice. Usually people wish some data about themselves were private while some other date may be public or is even wanted to be public for publicity reasons. There are different criteria which characterize the degree of anonymity of data. Given data can also be anonymized by different techniques in order to increase its degree of anonymity. In this paper we take a very simple "atomic" degree of anonymity as our starting place. We axiomatize these atoms and propose the investigation of first order logic based on these atoms. Considering the vast literature and the huge importance of anonymity our investigation may seem quite modest. However, it is about the logic of anonymity, not about how to secure, create or break anonymity.

cs.CR↗

arXiv2022

On highly equivalent non-isomorphic countable models of arithmetic and set theory

It is well-known that the first order Peano axioms PA have a continuum of non-isomorphic countable models. The question, how close to being isomorphic such countable models can be, seems to be less investigated. A measure of closeness to isomorphism of countable models is the length of back-and-forth sequences that can be established between them. We show that for every countable ordinal alpha there are countable non-isomorphic models of PA with a back-and-forth sequence of length alpha between them. This implies that the Scott height (or rank) of such models is bigger than $α$. We also prove the same result for models of ZFC.

math.LO↗

arXiv2022

Modular SAT-based techniques for reasoning tasks in team semantics

We study the complexity of reasoning tasks for logics in team semantics. Our main focus is on the data complexity of model checking but we also derive new results for logically defined counting and enumeration problems. Our approach is based on modular reductions of these problems into the corresponding problems of various classes of Boolean formulas. We illustrate our approach via several new tractability/intractability results.

cs.LO↗

arXiv2021

When cardinals determine the power set: inner models and Härtig quantifier logic

We make use of some observations on the core model, for example assuming $V=L [ E ]$, and that there is no inner model with a Woodin cardinal, and $M$ is an inner model with the same cardinals as $V$, then $V=M$. We conclude in this latter situation that "$x=\mathcal{P} ( y )$" is $Σ_{1} ( Card )$ where $Card$ is a predicate true of just the infinite cardinals. It is known that this implies the validities of second order logic are reducible to $V_I$ the set of validities of the Härtig quantifier logic. We draw some further conclusions on the Löwenheim number, $\ell_{I}$ of the latter logic: that if no $L[E]$ model has a cardinal strong up to an $\aleph$-fixed point, and $\ell_{I}$ is less than the least weakly inaccessible $δ$, then (i) $\ell_I$ is a limit of measurable cardinals of $K$; (ii) the Weak Covering Lemma holds at $δ$.

math.LO↗

arXiv2021

Diversity, Dependence and Independence

We introduce the concepts of dependence and independence in a very general framework. We use a concept of rank to study dependence and independence. By means of the rank we identify (total) dependence with inability to create more diversity, and (total) independence with the presence of maximum diversity. We show that our theory of dependence and independence covers a variety of dependence concepts, for example the seemingly unrelated concepts of linear dependence in algebra and dependence of variables in logic.

cs.LO↗

arXiv2021

Tractability Frontier of Data Complexity in Team Semantics

We study the data complexity of model-checking for logics with team semantics. We focus on dependence, inclusion, and independence logic formulas under both strict and lax team semantics. Our results delineate a clear tractability/intractability frontiers in data complexity of both quantifier-free and quantified formulas for each of the logics. For inclusion logic under the lax semantics, we reduce the model-checking problem to the satisfiability problem of so-called dual-Horn Boolean formulas. Via this reduction, we give an alternative proof for the known result that the data complexity of inclusion logic is in PTIME.

cs.LO↗

arXiv2021

Chain Logic and Shelah's Infinitary Logic

For a cardinal of the form $κ=\beth_κ$, Shelah's logic $L^1_κ$ has a characterisation as the maximal logic above $\bigcup_{λ<κ} L_{λ, ω}$ satisfying Strong Undefinability of Well Order (SUDWO). SUDWO is a strengthening of the Undefinability of Well Order (UDWO). We prove that if $κ$ is singular of countable cofinality, Karp's chain logic \cite{Karpintroduceschain} is above $L^1_κ$, while it is already known that it satisfies UDWO and Interpolation. Moreover, we show that in these circumstances, the chain logic is -- in a sense -- maximal among logics with chain models to satisfy UDWO. We then show that the chain logic gives a partial solution to Problem 1.4. from Shelah's \cite{Sh797}, which asked whether for $κ$ singular of countable cofinality there was a logic strictly between $ L_{κ^+, ω}$ and $L_{κ^+, κ^+}$ having Interpolation. We show that modulo accepting as the upper bound a model class of $L_{κ, κ}$, Karp's chain logic satisfies the required properties. In addition, we show that this chain logic is not $κ$-compact, a question that we have asked on various occasions. We contribue to the further development of chain logic by proving the Union Lemma and identifying the chain-independent fragment of the logic, showing that it still has considerable expressive power. In conclusion, we have shown that the simply defined chain logic emulates the logic $L^1_κ$ in satisfying Interpolation, undefinability of well-order and maximality with respect to it, and the Union Lemma. In addition it has a Completeness Theorem.

math.LO↗

arXiv2021

Logicality and Model Classes

We ask, when is a property of a model a logical property? According to the so-called Tarski-Sher criterion this is the case when the property is preserved by isomorphisms. We relate this to model-theoretic characteristics of abstract logics in which the model class is definable. This results in a graded concept of logicality in the terminology of Sagi. We investigate which characteristics of logics, such as variants of the Löwenheim-Skolem Theorem, Completeness Theorem, and absoluteness, are relevant from the logicality point of view, continuing earlier work by Bonnay, Feferman, and Sagi. We suggest that a logic is the more logical the closer it is to first order logic. We also offer a refinement of the result of McGee that logical properties of models can be expressed in $L_{\infty\infty}$ if the expression is allowed to depend on the cardinality of the model, based on replacing $L_{\infty\infty}$ by a ``tamer" logic.

math.LO↗

arXiv2020

Bounded Symbiosis and Upwards Reflection

Bagaria and Väänänen developed a framework for studying the large cardinal strength of downwards Löwenheim-Skolem theorems and related set theoretic reflection properties. The main tool was the notion of symbiosis, originally introduced by the third author. Symbiosis provides a way of relating model theoretic properties of strong logics to definability in set theory. In this paper we continue the systematic investigation of symbiosis and apply it to upwards Löwenheim-Skolem theorems and reflection principles. To achieve this, we need to adapt the notion of symbiosis to a new form, called bounded symbiosis. As one easy application, we obtain upper and lower bounds for the large cardinal strength of upwards Löwenheim-Skolem-type principles for second order logic.

math.LO↗

arXiv2020

Inner Models from Extended Logics: Part 1

If we replace first order logic by second order logic in the original definition of Gödel's inner model $L$, we obtain HOD. In this paper we consider inner models that arise if we replace first order logic by a logic that has some, but not all, of the strength of second order logic. Typical examples are the extensions of first order logic by generalized quantifiers, such as the Magidor-Malitz quantifier, the cofinality quantifier, or stationary logic. Our first set of results show that both $L$ and HOD manifest some amount of {\em formalism freeness} in the sense that they are not very sensitive to the choice of the underlying logic. Our second set of results shows that the cofinality quantifier gives rise to a new robust inner model between $L$ and HOD. We show, among other things, that assuming a proper class of Woodin cardinals the regular cardinals $>\aleph_1$ of $V$ are weakly compact in the inner model arising from the cofinality quantifier and the theory of that model is (set) forcing absolute and independent of the cofinality in question. We do not know whether this model satisfies the Continuum Hypothesis, assuming large cardinals, but we can show, assuming three Woodin cardinals and a measurable above them, that if the construction is relativized to a real, then on a cone of reals the Continuum Hypothesis is true in the relativized model.

math.LO↗

arXiv2020

Tracing Internal Categoricity

Informally speaking, the categoricity of an axiom system means that its non-logical symbols have only one possible interpretation that renders the axioms true. Although non-categoricity has become ubiquitous in the second half of the 20th century whether one looks at number theory, geometry or analysis, the first axiomatizations of such mathematical theories by Dedekind, Hilbert, Huntington, Peano and Veblen were indeed categorical. A common resolution of the difference between the earlier categorical axiomatizations and the more modern non-categorical axiomatizations is that the latter derive their non-categoricity from Skolem's Paradox and Gödel's Incompleteness Theorems, while the former, being second order, suffer from a heavy reliance on metatheory, where the Skolem-Gödel phenomenon re-emerges. Using second order meta-theory to avoid non-categoricity of the meta-theory would only seem to lead to an infinite regress. In this paper we maintain that internal categoricity breaks this traditional picture. It applies to both first and second order axiomatizations, although in the first order case we have so far only examples. It does not depend on the meta-theory in a way that would lead to an infinite regress. And it covers the classical categoricity results of early researchers. In the first order case it is weaker than categoricity itself, and in the second order case stronger. We give arguments suggesting that internal categoricity is the "right" concept of categoricity.

math.LO↗

arXiv2018

An extension of a theorem of Zermelo

We show that if (M,E,E') satisfies the first order Zermelo-Fraenkel axioms of set theory when the membership relation is E and also when the membership relation is E', and in both cases the formulas are allowed to contain both E and E', then (M,E) and (M,E') are isomorphic, and the isomorphism is definable in (M,E,E'). This extends Zermelo's 1930 theorem about second order ZFC.

math.LO↗

arXiv2016

Propositional Team Logics

We consider team semantics for propositional logic, continuing our previous work (Yang & Väänänen 2016). In team semantics the truth of a propositional formula is considered in a set of valuations, called a team, rather than in an individual valuation. This offers the possibility to give meaning to concepts such as dependence, independence and inclusion. We define an expressively maximal propositional team logic, called full propositional team logic. This requires going beyond the logical operations of classical propositional logic. We exhibit a hierarchy of logics between the smallest, viz. classical propositional logic, and the full propositional team logic. We characterize these different logics in several ways: first syntactically by their logical operations, and then semantically by the kind of sets of teams they are capable of defining. In several important cases we are able to find complete axiomatizations for these logics.

math.LO↗

arXiv2016

A dichotomy theorem for the generalized Baire space and elementary embeddability at uncountable cardinals

We consider the following dichotomy for $Σ^0_2$ finitary relations $R$ on analytic subsets of the generalized Baire space for $κ$: either all $R$-independent sets are of size at most $κ$, or there is a $κ$-perfect $R$-independent set. This dichotomy is the uncountable version of a result found in (W. Kubiś, Proc. Amer. Math. Soc. 131 (2003), no 2.:619--623) and in (S. Shelah, Fund. Math. 159 (1999), no. 1:1--50). We prove that the above statement holds assuming $\Diamond_κ$ and the set theoretical hypothesis $I^-(κ)$, which is the modification of the hypothesis $I(κ)$ suitable for limit cardinals. When $κ$ is inaccessible, or when $R$ is a closed binary relation, the assumption $\Diamond_κ$ is not needed. We obtain as a corollary the uncountable version of a result by G. Sági and the first author (Log. J. IGPL 20 (2012), no. 6:1064--1082) about the $κ$-sized models of a $Σ^1_1(L_{κ^+κ})$-sentence when considered up to isomorphism, or elementary embeddability, by elements of a $K_κ$ subset of ${}^κκ$. The role of elementary embeddings can be replaced by a more general notion that also includes embeddings, as well as the maps preserving $L_{λμ}$ for $ω\leqμ\leqλ\leqκ$ and the finite variable fragments of these logics.

math.LO↗

arXiv2016

A Logic for Arguing About Probabilities in Measure Teams

We use sets of assignments, a.k.a. teams, and measures on them to define probabilities of first-order formulas in given data. We then axiomatise first-order properties of such probabilities and prove a completeness theorem for our axiomatisation. We use the Hardy-Weinberg Principle of biology and the Bell's Inequalities of quantum physics as examples.

math.LO↗

arXiv2016

Dependence Logic in Pregeometries and $ω$-Stable Theories

We present a framework for studying the concept of independence in a general context covering database theory, algebra and model theory as special cases. We show that well-known axioms and rules of independence for making inferences concerning basic atomic independence statements are complete with respect to a variety of semantics. Our results show that the uses of independence concepts in as different areas as database theory, algebra and model theory, can be completely characterized by the same axioms. We also consider concepts related to independence, such as dependence.

math.LO↗