On Formalism Freeness : Implementing Gödel's 1946 Princeton Bicentennial Lecture

Peer reviewe

Indeterminateness and `The' Universe of Sets: Multiversism, Potentialism, and Pluralism

In this article, I survey some philosophical attitudes to talk concerning `the' universe of sets. I separate out four different strands of the debate, namely: (i) Universism, (ii) Multiversism, (iii) Potentialism, and (iv) Pluralism. I discuss standard arguments and counterarguments concerning the positions and some of the natural mathematical programmes that are suggested by the various views

Naming the largest number: Exploring the boundary between mathematics and the philosophy of mathematics

What is the largest number accessible to the human imagination? The question is neither entirely mathematical nor entirely philosophical. Mathematical formulations of the problem fall into two classes: those that fail to fully capture the spirit of the problem, and those that turn it back into a philosophical problem

Complete Deductive Systems for Probability Logic with Application in Harsanyi Type Spaces

Thesis (PhD) - Indiana University, Mathematics, 2007These days, the study of probabilistic systems is very popular not only in theoretical computer science but also in economics. There is a surprising concurrence between game theory and probabilistic programming. J.C. Harsanyi introduced the notion of type spaces to give an implicit description of beliefs in games with incomplete information played by Bayesian players. Type functions on type spaces are the same as the stochastic kernels that are used to interpret probabilistic programs. In addition to this semantic approach to interactive epistemology, a syntactic approach was proposed by R.J. Aumann. It is of foundational importance to develop a deductive logic for his probabilistic belief logic. In the first part of the dissertation, we develop a sound and complete probability logic $\Sigma_+$ for type spaces in a formal propositional language with operators $L_r^i$ which means ``the agent $i$'s belief is at least $r$" where the index $r$ is a rational number between 0 and 1. A crucial infinitary inference rule in the system $\Sigma_+$ captures the Archimedean property about indices. By the Fourier-Motzkin's elimination method in linear programming, we prove Professor Moss's conjecture that the infinitary rule can be replaced by a finitary one. More importantly, our proof of completeness is in keeping with the Henkin-Kripke style. Also we show through a probabilistic system with parameterized indices that it is decidable whether a formula $\phi$ is derived from the system $\Sigma_+$. The second part is on its strong completeness. It is well-known that $\Sigma_+$ is not strongly complete, i.e., a set of formulas in the language may be finitely satisfiable but not necessarily satisfiable. We show that even finitely satisfiable sets of formulas that are closed under the Archimedean rule are not satisfiable. From these results, we develop a theory about probability logic that is parallel to the relationship between explicit and implicit descriptions of belief types in game theory. Moreover, we use a linear system about probabilities over trees to prove that there is no strong completeness even for probability logic with finite indices. We conclude that the lack of strong completeness does not depend on the non-Archimedean property in indices but rather on the use of explicit probabilities in the syntax. We show the completeness and some properties of the probability logic for Harsanyi type spaces. By adding knowledge operators to our languages, we devise a sound and complete axiomatization for Aumann's semantic knowledge-belief systems. Its applications in labeled Markovian processes and semantics for programs are also discussed

Large Cardinals

Infinite sets are a fundamental object of modern mathematics. Surprisingly, the existence of infinite sets cannot be proven within mathematics. Their existence, or even the consistency of their possible existence, must be justified extra-mathematically or taken as an article of faith. We describe here several varieties of large infinite set that have a similar status in mathematics to that of infinite sets, i.e. their existence cannot be proven, but they seem both reasonable and useful. These large sets are known as large cardinals. We focus on two types of large cardinal: inaccessible cardinals and measurable cardinals. Assuming the existence of a measurable cardinal allows us to disprove a questionable statement known as the Axiom of Constructibility (V=L)

Flexibility in Ceteris Paribus Reasoning

Ceteris Paribus clauses in reasoning are used to allow for defeaters of norms, rules or laws, such as in von Wright’s example “I prefer my raincoat over my umbrella, everything else being equal”. In earlier work, a logical analysis is offered in which sets of formulas Γ, embedded in modal operators, provide necessary and sufficient conditions for things to be equal in ceteris paribus clauses. For most laws, the set of things allowed to vary is small, often finite, and so Γ is typically infinite. Yet the axiomatisation they provide is restricted to the special and atypical case in which Γ is finite. We address this problem by being more flexible about ceteris paribus conditions, in two ways. The first is to offer an alternative, slightly more general semantics, in which the set of formulas only give necessary but not (necessarily) sufficient conditions. This permits a simple axiomatisation

The Structure of Models of Second-order Set Theories

This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The approach is to understand second-order set theory by studying the structure of models of second-order set theories. The main results are the following, organized by chapter. First, I investigate the poset of T-realizations of a fixed countable model of ZFC, where T is a reasonable second-order set theory such as GBC or KM, showing that it has a rich structure. In particular, every countable partial order embeds into this structure. Moreover, we can arrange so that these embedding preserve the existence/nonexistence of upper bounds, at least for finite partial orders. Second I generalize some constructions of Marek and Mostowski from KM to weaker theories. They showed that every model of KM plus the Class Collection schema “unrolls” to a model of ZFC− with a largest cardinal. I calculate the theories of the unrolling for a variety of second-order set theories, going as weak as GBC + ETR. I also show that being T-realizable goes down to submodels for a broad selection of second-order set theories T. Third, I show that there is a hierarchy of transfinite recursion principles ranging in strength from GBC to KM. This hierarchy is ordered first by the complexity of the properties allowed in the recursions and second by the allowed heights of the recursions. Fourth, I investigate the question of which second-order set theories have least models. I show that strong theories—such as KM or Π11-CA—do not have least transitive models while weaker theories—from GBC to GBC + ETROrd —do have least transitive models

The first-order logic of CZF is intuitionistic first-order logic

We prove that the first-order logic of CZF is intuitionistic first-order logic. To do so, we introduce a new model of transfinite computation (Set Register Machines) and combine the resulting notion of realisability with Beth semantics. On the way, we also show that the propositional admissible rules of CZF are exactly those of intuitionistic propositional logic.Comment: Revised version, more precise title, 20 page