The received notion of axiomatic theory, which stems from Hilbert, is that of set T of propositions (either contentful or non-interpreted aka propositional forms) with subset A of axioms provided with a notion of consequence, which generates T from A in the obvious way. I argue that this standard notion is too narrow for being an adequate theoretical model of many mathematical theories; the class of such counter-examples is apparently very large and it includes such different theories as the geometrical theory of Euclid’s Elements, Book 1, and the more recent Homotopy type theory. In order to fix this problem I introduce a more general notion of theory, which uses typing and a generalized notion of consequence applicable also to objects of other types than propositions. I call such a theory constructive axiomatic theory and show that this particular conception of being constructive indeed captures many important ideas concerning the mathematical constructivity found in the earlier literature from Hilbert to Kolmogorov to Martin-Lof. Finally I provide an epistemological argument intended to show that the notion of constructive axiomatic theory is more apt to be useful in natural sciences and other empirical contexts than the standard notion. Disclaimer: The notion of constructive axiomatic theory is not my invention. The idea and its technical implementation are found in Martin-Lof ‘s constructive type theory if not already in Euclid. My aim is to make this notion explicit and introduce it into the continuing discussions on axiomatic method and mathematical and logical constructivity.

This talk — a mix of mathematics and philosophy — concerns the extent to which we may infer definiteness of truth in a mathematical context from definiteness of the underlying objects and structure of that context. The philosophical analysis is based in part on the mathematical observation that the satisfaction relation for model-theoretic truth is less absolute than often supposed.  Specifically, two models of set theory can have the same natural numbers and the same structure of arithmetic in common, yet disagree about whether a particular arithmetic sentence is true in that structure. In other words, two models can have the same arithmetic objects and the same formulas and sentences in the language of arithmetic, yet disagree on their corresponding theories of truth for those objects. Similarly, two models of set theory can have the same natural numbers, the same arithmetic structure, and the same arithmetic truth, yet disagree on their truths-about-truth, and so on at any desired level of the iterated truth-predicate hierarchy.  These mathematical observations, for which I shall strive to give a very gentle proof in the talk (using only elementary classical methods), suggest that a philosophical commitment to the determinate nature of the theory of truth for a structure cannot be seen as a consequence solely of the determinateness of the structure in which that truth resides. The determinate nature of arithmetic truth, for example, is not a consequence of the determinate nature of the arithmetic structure N = {0,1,2,…} itself, but rather seems to be an additional higher-order commitment requiring its own analysis and justification.

This work is based on my recent paper, Satisfaction is not absolute, joint with Ruizhi Yang (Fudan University, Shanghai).

Long before Kripke semantics became standard in modal logic, Tarski showed us that the basic propositional modal language can be interpreted in topological spaces. In Tarski’s semantics for the modal logic $S4$, each propositional variable is evaluated to an arbitrary subset of a fixed topological space. I develop a related, measure theoretic semantics, in which modal formulas are interpreted in the Lebesgue measure algebra, or algebra of Borel subsets of the real interval $[0,1]$, modulo sets of measure zero. This semantics was introduced by Dana Scott in the last several years. I discuss some of my own completeness results, and ways of extending the semantics to more complex modal languages.

I will discuss a number of theorems showing that the satisfaction relation of first-order logic is less absolute than might have been supposed. Two models of set theory $M_1$ and $M_2$, for example, can agree on their natural numbers $langlemathbb{N},{+},{cdot},0,1,{lt}rangle^{M_1}=langlemathbb{N},{+},{cdot},0,1,{lt}rangle^{M_2}$, yet disagree on arithmetic truth: they have a sentence $sigma$ in the language of arithmetic that $M_1$ thinks is true in the natural numbers, yet $M_2$ thinks $negsigma$ there. Two models of set theory can agree on the natural numbers $mathbb{N}$ and on the reals $mathbb{R}$, yet disagree on projective truth. Two models of set theory can have the same natural numbers and have a computable linear order in common, yet disagree about whether this order is well-ordered. Two models of set theory can have a transitive rank initial segment $V_delta$ in common, yet disagree about whether this $V_delta$ is a model of ZFC. The theorems are proved with elementary classical methods.

This is joint work with Ruizhi Yang (Fudan University, Shanghai). We argue, on the basis of these mathematical results, that the definiteness of truth in a structure, such as with arithmetic truth in the standard model of arithmetic, cannot arise solely from the definiteness of the structure itself in which that truth resides; rather, it must be seen as a separate, higher-order ontological commitment.

This will be a talk on April 30, 2013 for a joint meeting of the Yeshiva University Mathematics Club and the Yeshiva University Philosophy Club. I will give a general introduction to the theory of infinite games, suitable for mathematicians and philosophers. What does it mean to play an infinitely long game? What does it mean to have a winning strategy for such a game? Is there any reason to think that every game should have a winning strategy for one player or another? Could there be a game, such that neither player has a way to force a win? Must every computable game have a computable winning strategy? I will present several game paradoxes and example infinitary games, including an infinitary version of the game of Nim, and several examples from infinite chess.

I shall give a summary account of some current issues in the philosophy of set theory, specifically, the debate on pluralism and the question of the determinateness of set-theoretical and mathematical truth. The traditional Platonist view in set theory, what I call the universe view, holds that there is an absolute background concept of set and a corresponding absolute background set-theoretic universe in which every set-theoretic assertion has a final, definitive truth value. What I would like to do is to tease apart two often-blurred aspects of this perspective, namely, to separate the claim that the set-theoretic universe has a real mathematical existence from the claim that it is unique. A competing view, which I call the multiverse view, accepts the former claim and rejects the latter, by holding that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe, and a corresponding pluralism of set-theoretic truths. After framing the dispute, I shall argue that the multiverse position explains our experience with the enormous diversity of set-theoretic possibility, a phenomenon that is one of the central set-theoretic discoveries of the past fifty years and one which challenges the universe view. In particular, I shall argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for.