  This paper draws on a number of Roger Penrose's 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.
