arXiv subjects
Publications and source records attributed to 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 chains and antichains in posets and Menger's theorem about disjoint paths in directed graphs.
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 triangle-free operator systems.
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 Tarski's "convention T", but compositionality is automatic. The correct, non-paradoxical form of Frege's Basic Law V is given. This paper is a (very) condensed version of the account of truth advanced in my recent book *Constructive Countablism*.
Hierarchies of non self-applicative truth predicates take us beyond Gamma_0.
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 separable metric space, and the utility functions may be unbounded. The main ingredient of our results is a novel (behavioral) axiom on the underlying preference relations which is satisfied by virtually all stochastic orders. The proof of the main representation theorem is built on the fact that the completion of the Kantorovich-Rubinstein space is the canonical predual of the Banach space of Lipschitz functions that vanish at a fixed point. Two applications are given, one to the theory of non-expected utility theory, and the other to the theory of decision-making under uncertainty.
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 prove matrix analogs of the theorems of Dilworth and Mirsky for finite posets. Some partial results are obtained in the infinite dimensional case.
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 fallacy, first noted in [N. Weaver, Predicativity beyond Gamma_0, arXiv:math/0509244], and some of its consequences.
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" is a rank k orthogonal projection for which dim(PVP) = 1. We give upper and lower bounds both for the largest dimension of V which would ensure the existence of a quantum k-anticlique, and for the smallest dimension of V which would ensure the existence of a quantum k-clique.
The liar paradox is widely seen as not a serious problem. I try to explain why this view is mistaken.
For any metric space X, the predual of Lip(X) is unique. A previous version of this manuscript, which is also the published version (Math. Prof. Cambridge Philos. Soc. 165 (2018), 467-473), additionally stated "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." However, the proof of a crucial lemma, Lemma 3.1 in the previous version, was faulty. The error in that proof lay in assuming that the limit, for the topology induced by W, of a net in the unit ball would have to lie in the unit ball. But we do not know that W is 1-norming. This error was pointed out by Manuel Gonzalez, as relayed to me by Ruben Medina. The reduction from "complete and convex" to "finite diameter" is still valid, and is retained in the present version.
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 cardinal conditions. For instance, there is a singular countably additive pure state on $B(l^2(\kappa))$ if and only if $\kappa$ is Ulam measurable, and there is a singular ${<}\,\kappa$-additive pure state on $B(l^2(\kappa))$ if and only if $\kappa$ is measurable. The proofs make use of Farah and Weaver's theory of quantum filters. Applications to Ueda's peak set theorem for von Neumann algebras are discussed in the final section.
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) = 1 or k^2.
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 interpret the Knill-Laflamme error-correction conditions [E. Knill and R. Laflamme, Theory of quantum error-correcting codes, Phys. Rev. A 55 (1997), 900-911] in terms of graph-theoretic independence, to give intrinsic characterizations of Stahlke's noncommutative graph homomorphisms [D. Stahlke, Quantum source-channel coding and non-commutative graph theory, arXiv:1405.5254] and Duan, Severini, and Winter's noncommutative bipartite graphs [R. Duan, S. Severini, and A. Winter, Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovasz number, IEEE Trans. Inform. Theory 59 (2013), 1164-1174], and to realize the noncommutative confusability graph associated to a quantum channel as the pullback of a diagonal relation. Our framework includes as special cases not only purely classical and purely quantum information theory, but also the "mixed" setting which arises in quantum systems obeying superselection rules. Thus we are able to define noncommutative confusability graphs, give error correction conditions, and so on, for such systems. This could have practical value, as superselection constraints on information encoding can be physically realistic.
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 subspace whose dimension is small compared to |G| = dim(l^2(G)) tends to be far away from standard subspaces. In particular, the recent positive solution of the Kadison-Singer problem shows that from within any Fourier subspace whose dimension is small compared to |G| there is standard subspace which is essentially indistinguishable from its orthogonal complement.
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 rational agents who are licensed to act under precisely defined conditions.
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 also investigate the problem of interpreting arbitrary formal systems in systems which include a provability operator.
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 suffices to resolve the difficulty. We are then able to present a formal system for reasoning about concepts. We prove that this system is consistent and we investigate the extent to which it is able to interpret set theoretic and number theoretic systems of a more standard type.