In the 90s, Kirchberg proved that the famous Connes Embedding Problem is equivalent to the statement that every C*-algebra has the QWEP (short for being a quotient of a C*-algebra with the weak expectation property). This equivalence was instrumental in the recent refutation of the Connes Embedding Problem via the quantum complexity result known as MIP*=RE. In an earlier work, I showed that the QWEP is an axiomatizable property via a soft semantic argument, leaving open the question of whether or not concrete axioms could be given for this class. In this talk, I will describe joint work with Jan Arulseelan and Bradd Hart, where we show that the class of QWEP C*-algebras does not admit an effective axiomatization. The proof uses the model-theoretic approach to the refutation of the Connes Embedding Problem from MIP*=RE due to Hart and myself. This aforementioned result involved showing that the hyperfinite II$_1$ factor $\mathcal{R}$ has an undecidable universal theory and in this talk I will also discuss how analogous results can be shown for the hyperfinite III$_1$ factor $\mathcal{R}_\infty$ and the hyperfinite III$_\lambda$ factor $\mathcal{R}_\lambda$ for $0< \lambda< 1$ (also joint with Arulseelan and Hart). In the final part of this talk, I will discuss the notion of the Tsirelson property for a C*-algebra and show that the class of such algebras, while axiomatizable, also does not admit an effective axiomatization. This latter part of the talk represents joint work with Hart.