Quantum Set Theory Extending the Standard Probabilistic Interpretation of Quantum Theory (Extended Abstract)

The notion of equality between two observables will play many important roles in foundations of quantum theory. However, the standard probabilistic interpretation based on the conventional Born formula does not give the probability of equality relation for a pair of arbitrary observables, since the Born formula gives the probability distribution only for a commuting family of observables. In this paper, quantum set theory developed by Takeuti and the present author is used to systematically extend the probabilistic interpretation of quantum theory to define the probability of equality relation for a pair of arbitrary observables. Applications of this new interpretation to measurement theory are discussed briefly.Comment: In Proceedings QPL 2014, arXiv:1412.810

Countable ordinals and the analytical hierarchy, I

The following results are proved, using the axiom of Projective Determinacy: (i) For n ≥ 1, every II(1/2n+1) set of countable ordinals contains a Δ(1/2n+1) ordinal, (ii) For n ≥ 1, the set of reals Δ(1/2n) in an ordinal is equal to the largest countable Σ(1/2n) set and (iii) Every real is Δ(1/n) inside some transitive model of set theory if and only if n ≥ 4

Structural Relativity and Informal Rigour

Informal rigour is the process by which we come to understand particular mathematical structures and then manifest this rigour through axiomatisations. Structural relativity is the idea that the kinds of structures we isolate are dependent upon the logic we employ. We bring together these ideas by considering the level of informal rigour exhibited by our set-theoretic discourse, and argue that different foundational programmes should countenance different underlying logics (intermediate between first- and second-order) for formulating set theory. By bringing considerations of perturbations in modal space to bear on the debate, we will suggest that a promising option for representing current set-theoretic thought is given by formulating set theory using quasi-weak second-order logic. These observations indicate that the usual division of structures into \particular (e.g. the natural number structure) and general (e.g. the group structure) is perhaps too coarse grained; we should also make a distinction between intentionally and unintentionally general structures

History of consumer demand theory 1871-1971: A Neo-Kantian rational reconstruction

This paper examines the history of the neoclassical theory of consumer demand from 1871 to 1971 by bringing into play the knowledge theory of the Marburg School, a Neo-Kantian philosophical movement. The work aims to show the usefulness of a Marburg-inspired epistemology in rationalizing the development of consumer analysis and, more generally, to understand the principles that regulate the process of knowing in neoclassical economics.Consumer Theory, Demand Theory, Utility Theory, Neo- Kantianism, Marburg School

On Martin's Pointed Tree Theorem

We investigate the reverse mathematics strength of Martin's pointed tree theorem (MPT) and one of its variants, weak Martin's pointed tree theorem (wMPT)

The Power of Naive Truth

While non-classical theories of truth that take truth to be transparent have some obvious advantages over any classical theory that evidently must take it as non-transparent, several authors have recently argued that there's also a big disadvantage of non-classical theories as compared to their “external” classical counterparts: proof-theoretic strength. While conceding the relevance of this, the paper argues that there is a natural way to beef up extant internal theories so as to remove their proof-theoretic disadvantage. It is suggested that the resulting internal theories should seem preferable to their external counterparts