We survey the use of extra-set-theoretic hypotheses, mainly the continuum hypothesis, in the C*-algebra literature. The Calkin algebra emerges as a basic object of interest.

We present a characterization of supercompactness measures for $\omega_1$ in L($\mathbb{R}$), and of countable products of such measures, using inner models. We give two applications of this characterization, the first obtaining the consistency of $\delta_3^1=\omega_2$ with $\mathsf{ZFC+AD}^{L\mathbb{R}}$, and the second proving the uniqueness of the supercompactness measure over $\mathcal{P}_{\omega_l }(\lambda)$ in L ($\mathbb{R}$) for $\lambda > \delta_1^2$.

We introduce a nonstandard arithmetic NQA⁻ based on the theory developed by R. Chuaqui and P. Suppes in [2] (we will denote it by NQA⁺), with a weakened external open minimization schema. A finitary consistency proof for NQA⁻ formalizable in PRA is presented. We also show interesting facts about the strength of the theories NQA⁻ and NQA⁺; NQA⁻ is mutually interpretable with IΔ₀+EXP, and on the other hand, NQA⁺ interprets the theories IΣ₁ and WKL₀.