This paper draws on a number of Roger Penroses ideas - including the non-Hamiltonian phase-space flow of the Hawking Box, Conformal Cyclic Cosmology, non-computability and gravitationally induced quantum state reduction - in order to propose a radically unconventional approach to quantum gravity: Invariant Set Theory (IST). In IST, the fundamental laws of physics describe the geometry of the phase portrait of the universe as a whole: quantum process are associated with fine-scale fractal geometry, gravitational process with larger-scale heterogeneous geometry. With this, it becomes possible to explain the experimental violation of Bell Inequalities without having to abandon key ingredients of general relativity: determinism and local causality. Ensembles in IST can be described by complex Hilbert states over a finite set $mathbb C_p$ of complex numbers, where $p$ is a large finite integer. The quantum mechanics of finite-dimensional Hilbert spaces is emergent as a singular limit when $p rightarrow infty$. A small modification to the field equations of general relativity is proposed to make it consistent with IST.
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