What is the correct noncommutative generalization of the functor $C_0(X) \mapsto \ell^\infty(X)$ for locally compact Hausdorff $X$ having a countable basis? Making the ansatz $K(\ell^2) \mapsto B(\ell^2)$, we expect that every unital $*$-homomorphism $C(\mathbb T) \rightarrow B(\ell^2)$ extend canonically to a unital $*$-homomorphism $\ell^\infty(\mathbb T) \rightarrow B(\ell^2)$. Thus, we expect to extend the continuous functional calculus for a unitary operator on $\ell^2$ to all bounded complex-valued functions. Therefore, we work in a model of set theory where every set of real numbers is Lebesgue measurable; we must assume the consistency of an inaccessible cardinal in order to do so. The axiom of choice necessarily fails in such a model, but our model is carefully chosen to enable the verification of many familiar theorems via a scrutinization of their statements rather than their proofs. This technique significantly lowers the cost of doing interesting mathematics in this unfamiliar setting, and it is explained in detail. By analogy with the ultraweak topology, we define the continuum-weak topology on bounded operators to be the topology given by functionals of the form $x \mapsto \int_0^1 <\eta_t| x \xi_t>\, dt$. We then define a V*-algebra to be a C*-algebra of bounded operators that is closed in the continuum-weak topology. Every C*-algebra $A$ has an enveloping V*-algebra $V^*(A)$, and if $X$ is a locally compact Hausdorff space with a countable basis, then $V^*(C_0(X)) \cong \ell^\infty(X)$. More generally, if $A$ is any separable C*-algebra of type I, then $V^*(A)$ is canonically isomorphic to an $\ell^\infty$-direct sum of type I factors, with one summand for each irreducible representation of $A$. The self-adjoint part of any unital separable C*-algebra is isomorphic to the Banach space of strongly affine real-valued functions on its state space.

We investigate the position that foundational theories should be modelled on ordinary computability, with axioms and theorems presented as sequents of Σ formulas. We define a proof from such a theory to be a finite sequence of Σ formulas, with an axiom of the theory applied at each step. We obtain a complete system of logical axioms for this style of proof. In such a proof, free variables are not implicitly universally quantified, but rather behave like parameters, so each formula may be computationally verified, notionally in an infinitary setting, and literally in a finitary one. We show that such a foundational theory may establish the semantic properties of the Σ truth predicate, and may additionally establish that the truth of the assumption of any of its axioms implies the truth of the conclusion, but nevertheless that theory may be strong enough to formalize standard foundational theories in both the finitary and infinitary settings. We also show that the Σ truth predicate may be extended to intuitionistic sentences in a way that respects the connectives permitted in Σ formulas, by appealing to the constructive properties of intuitionstic deduction. Finally, we propose two completeness principles for the universe of pure sets, and show that they are equivalent.

We develop the viewpoint that the opposite of the category of W*-algebras and unital normal *-homomorphisms is analogous to the category of sets and functions. For each pair of W*-algebras, we construct their free exponential, which in the context of this analogy corresponds to the collection of functions from one of these W*-algebras to the other. We also show that every unital normal completely positive map between W*-algebras arises naturally from a normal state on their free exponential.