Nik Weaver
We review known linear and matrix generalizations of Hall's classic ``marriage theorem'' and K\H{o}nig's theorem on partial matchings in bipartite graphs, and relate them to linear and matrix generalizations of Dilworth's theorem about…
We propose axioms governing the interaction of constructive assertibility and meaningfulness predicates with a self-applicative truth predicate characterized by the T-scheme, and we prove the consistency of the resulting formal system.
We introduce notions of being "triangle-free" and "strongly triangle-free" for operator systems in M_n(C) considered as quantum graphs. Several examples and non-examples are discussed. We provide a complete characterization of strongly…
I outline a new theory of truth that resolves the classical and constructive versions of the liar paradox. The theory features a provably consistent axiomatization of a global self-applicative truth predicate. Truth is defined using…
Hierarchies of non self-applicative truth predicates take us beyond Gamma_0.
We introduce a notion of ``hereditarily antisymmetric'' operator algebras and prove a structure theorem for them in finite dimensions. We also characterize those operator algebras in finite dimensions which can be made upper triangular and…
We obtain variants of the classical von Neumann-Morgenstern expected utility theorem, with and without the completeness axiom, in which the derived Bernoulli utility functions are Lipschitz. The prize space in these results is an arbitrary…
Confusion over the predicativist conception of well-ordering pervades the literature and is responsible for widespread fundamental misconceptions about the nature of predicative reasoning. This short note aims to explain the principal…
Let V be a linear subspace of M_n(C) which contains the identity matrix and is stable under Hermitian transpose. A "quantum k-clique" for V is a rank k orthogonal projection P in M_n(C) for which dim(PVP) = k^2, and a "quantum k-anticlique"…
The "noncommutative graphs" which arise in quantum error correction are a special case of the quantum relations introduced in [N. Weaver, Quantum relations, Mem. Amer. Math. Soc. 215 (2012), v-vi, 81-140]. We use this perspective to…
Marcus, Spielman, and Srivastava recently solved the Kadison-Singer problem by showing that if u_1, ..., u_m are column vectors in C^d such that \sum u_iu_i^* = I, then a set of indices S \subseteq {1, ..., m} can be chosen so that \sum_{i…
The liar paradox is widely seen as not a serious problem. I try to explain why this view is mistaken.
We investigate states on von Neumann algebras which are not normal but enjoy various forms of infinite additivity, and show that these exist on $B(H)$ if and only if the cardinality of an orthonormal basis of $H$ satisfies various large…
For any metric space X, the predual of Lip(X) is unique. If X has finite diameter or is complete and convex --- in particular, if it is a Banach space --- then the predual of Lip_0(X) is unique.
Let V be a linear subspace of M_n(C) which contains the identity matrix and is stable under the formation of Hermitian adjoints. We prove that if n is sufficiently large then there exists a rank k orthogonal projection P such that dim(PVP)…
Let G be a finite abelian group. We examine the discrepancy between subspaces of l^2(G) which are diagonalized in the standard basis and subspaces which are diagonalized in the dual Fourier basis. The general principle is that a Fourier…
We consider extensions of Peano arithmetic which include an assertibility predicate. Any such system which is arithmetically sound effectively verifies its own soundness. This leads to the resolution of a range of paradoxes involving…
We find that second order quantification is problematic when a quantified concept variable is supposed to function predicatively. This issue is analyzed and it is shown that a constructive interpretation of the falling under relation…
We analyze the informal semantic conception of proof and axiomatize the proof relation and the provability operator. A self referential propositional calculus which admits provable liar type sentences is introduced and proven consistent. We…
We apply Srivastava's spectral sparsification technique to a vector balancing version of the Kadison-Singer problem. The result is a one-sided version of the conjectured solution.