ZFC has sentences that quantify over all sets or all ordinals, without restriction. Some have argued that sentences of this kind lack a determinate meaning. We propose a set theory called TOPS, using Natural Deduction, that avoids this problem by speaking only about particular sets.

We solve a longstanding problem by providing a denotational model for nondeterministic programs that identifies two programs iff they have the same range of possible behaviours. We discuss the difficulties with traditional approaches, where divergence is bottom or where a term denotes a function from a set of environments. We see that making forcing explicit, in the manner of game semantics, allows us to avoid these problems.We begin by modelling a first-order language with sequential I/O and unbounded nondeterminism. Then we (...) extend the model to a calculus with higher-order and recursive types, by adapting standard game semantics. Traditional adequacy proofs using logical relations are not applicable, so we use instead a novel hiding and unhiding argument. (shrink)

"The Quantum Revelation is mind-blowing." --Sting To say that quantum physics is the greatest scientific discovery of all time is not an exaggeration. In their discovery of the quantum realm, the physics community stumbled upon a genuine multifaceted revelation which can be likened to a profound spiritual treasure--a heretofore undreamed of creative power--hidden within our own mind. Quantum physics unequivocally points out that the study of the universe and the study of consciousness are inseparably linked, which is to say that (...) ultimate progress in the one will be impossible without progress in the other. Einstein declared that what it reveals is so immensely important that "it should be everyone's concern." Yet few of us in the general public truly understand how the game-changing discoveries of the past century not only relate to our day-to-day lives, but also give us insights into the nature of reality and our place within it. Written for readers with no physics background, Paul Levy's latest book, The Quantum Revelation: A Radical Synthesis of Science and Spirituality is for those who have heard that quantum physics is a fascinating subject but don't quite understand how or why. Levy contemplates the deeper philosophical underpinnings of quantum physics, exploring the fundamental questions it provokes: What does it mean that quantum theory has discovered that there is no such thing as "objective reality?" How are we participating--via our consciousness--in creating our experience of a reality that quantum theory itself describes as "dreamlike?" What are the implications for us in our day-to-day lives that--as quantum theory reveals--what we call reality is more like a dream that we had previously imagined? The Quantum Revelation is unique in how it synthesizes science and spirituality so as to reveal and explore the dreamlike nature of reality. It is a book not just for people interested in quantum theory, but for anyone who is interested in waking up and dreaming lucidly, be it in our night dreams or our waking life. (shrink)

This paper argues that, insofar as we doubt the bivalence of the Continuum Hypothesis or the truth of the Axiom of Choice, we should also doubt the consistency of third-order arithmetic, both the classical and intuitionistic versions. -/- Underlying this argument is the following philosophical view. Mathematical belief springs from certain intuitions, each of which can be either accepted or doubted in its entirety, but not half-accepted. Therefore, our beliefs about reality, bivalence, choice and consistency should all be aligned.