Troubles with (the concept of) truth in mathematics

In the paper the problem of definability and undefinability of the concept of satisfaction and truth is considered. Connections between satisfaction and truth on the one hand and consistency of certain systems of omega-logic and transfinite induction on the other are indicated

Abstracta and Possibilia: Modal Foundations of Mathematical Platonism

This paper aims to provide modal foundations for mathematical platonism. I examine Hale and Wright's (2009) objections to the merits and need, in the defense of mathematical platonism and its epistemology, of the thesis of Necessitism. In response to Hale and Wright's objections to the role of epistemic and metaphysical modalities in providing justification for both the truth of abstraction principles and the success of mathematical predicate reference, I examine the Necessitist commitments of the abundant conception of properties endorsed by Hale and Wright and examined in Hale (2013); and demonstrate how a two-dimensional approach to the epistemology of mathematics is consistent with Hale and Wright's notion of there being non-evidential epistemic entitlement rationally to trust that abstraction principles are true. A choice point that I flag is that between availing of intensional or hyperintensional semantics. The hyperintensional semantic approach that I advance is a topic-sensitive epistemic two-dimensional truthmaker semantics. Epistemic and metaphysical states and possibilities may thus be shown to play a constitutive role in vindicating the reality of mathematical objects and truth, and in providing a conceivability-based route to the truth of abstraction principles as well as other axioms and propositions in mathematics

Polylogarithmic Cuts in Models of V^0

We study initial cuts of models of weak two-sorted Bounded Arithmetics with respect to the strength of their theories and show that these theories are stronger than the original one. More explicitly we will see that polylogarithmic cuts of models of $\mathbf{V}^0$ are models of $\mathbf{VNC}^1$ by formalizing a proof of Nepomnjascij's Theorem in such cuts. This is a strengthening of a result by Paris and Wilkie. We can then exploit our result in Proof Complexity to observe that Frege proof systems can be sub exponentially simulated by bounded depth Frege proof systems. This result has recently been obtained by Filmus, Pitassi and Santhanam in a direct proof. As an interesting observation we also obtain an average case separation of Resolution from AC0-Frege by applying a recent result with Tzameret.Comment: 16 page

Varieties of truth definitions

We study the structure of the partial order induced by the definability relation on definitions of truth for the language of arithmetic. Formally, a definition of truth is any sentence $\alpha$ which extends a weak arithmetical theory (which we take to be EA) such that for some formula $\Theta$ and any arithmetical sentence $\varphi$, $\Theta(\ulcorner\varphi\urcorner)\equiv \varphi$ is provable in $\alpha$. We say that a sentence $\beta$ is definable in a sentence $\alpha$, if there exists an unrelativized translation from the language of $\beta$ to the language of $\alpha$ which is identity on the arithmetical symbols and such that the translation of $\beta$ is provable in $\alpha$. Our main result is that the structure consisting of truth definitions which are conservative over the basic arithmetical theory forms a countable universal distributive lattice. Additionally, we generalize the result of Pakhomov and Visser showing that the set of (G\"odel codes of) definitions of truth is not $\Sigma_2$-definable in the standard model of arithmetic. We conclude by remarking that no $\Sigma_2$-sentence, satisfying certain further natural conditions, can be a definition of truth for the language of arithmetic

Dominating the Erdos-Moser theorem in reverse mathematics

The Erdos-Moser theorem (EM) states that every infinite tournament has an infinite transitive subtournament. This principle plays an important role in the understanding of the computational strength of Ramsey's theorem for pairs (RT^2_2) by providing an alternate proof of RT^2_2 in terms of EM and the ascending descending sequence principle (ADS). In this paper, we study the computational weakness of EM and construct a standard model (omega-model) of simultaneously EM, weak K\"onig's lemma and the cohesiveness principle, which is not a model of the atomic model theorem. This separation answers a question of Hirschfeldt, Shore and Slaman, and shows that the weakness of the Erdos-Moser theorem goes beyond the separation of EM from ADS proven by Lerman, Solomon and Towsner.Comment: 36 page

Topics in arithmetic and determinacy

This thesis is about Arithmetical Determinacy. Loosely, this is the problem of whether every question in arithmetic has a determinate answer. In this work I discuss how to exactly understand the concept of determinacy, I criticise arguments for and against the claim that arithmetic is determinate, and examine how questions about determinacy may be applied to other debates in the philosophy of mathematics. Chapter 1 isolates different ways of understanding the problem of arithmetical determinacy. Chapter 2 turns to mathematical structuralism and explains how popular computability constraints thought to determine the reference of our arithmetical vocabulary are actually unsuccessful in securing determinacy. Chapter 3 criticises an interesting idea for securing determinacy via our experience with supertasks. Chapter 4 explores the phenomenon of mutually inconsistent satisfaction classes and motivates a new account of determinacy in terms of sentences possessing non-classical truth-values. Chapter 5 defends strict finitism, framing some objections against the view in terms of the concept of arithmetical determinacy