A new proof of James' sup theorem.
We provide a new proof of James' sup theorem for (non necessarily separable) Banach spaces. One of the ingredients is the following generalization of a theorem of Hagler and Johnson: .
Linear forms and axioms of choice
We work in set-theory without choice ZF. Given a commutative field , we consider the statement : “On every non null -vector space there exists a non-null linear form.” We investigate various statements which are equivalent to in ZF. Denoting by the two-element field, we deduce that implies the axiom of choice for pairs. We also deduce that implies the axiom of choice for linearly ordered sets isomorphic with .
Linear extenders and the Axiom of Choice
In set theory without the Axiom of Choice ZF, we prove that for every commutative field , the following statement : “On every non null -vector space, there exists a non null linear form” implies the existence of a “-linear extender” on every vector subspace of a -vector space. This solves a question raised in Morillon M., Linear forms and axioms of choice, Comment. Math. Univ. Carolin. 50 (2009), no. 3, 421-431. In the second part of the paper, we generalize our results in the case of spherically...
Hyperplanes in matroids and the axiom of choice
We show that in set theory without the axiom of choice ZF, the statement sH: “Every proper closed subset of a finitary matroid is the intersection of hyperplanes including it” implies AC, the axiom of choice for (nonempty) finite sets. We also provide an equivalent of the statement AC in terms of “graphic” matroids. Several open questions stay open in ZF, for example: does sH imply the axiom of choice?
Three generators for minimal writing-space computations
Three generators for minimal writing-space computations
We construct, for each integer , three functions from {0,1} to {0,1} such that any boolean mapping from {0,1} to {0,1} can be computed with a finite sequence of assignations only using the input variables and those three functions.
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