I have found a paper that seems to cover most of my thoughts about the
arithmetic body problem:
Models of axiomatic theories admitting automorphisms
by A. Ehrenfeucht A. Mostowski
More on related concepts are found in the Vaught conjecture:
"The topological Vaught conjecture is the statement that whenever a
Polish group acts continuously on a Polish space, there are either
countably many orbits or continuum many orbits. The topological Vaught
conjecture is more general than the original Vaught conjecture: Given a
countable language we can form the space of all structures on the
natural numbers for that language. If we equip this with the topology
generated by first order formulas, then it is known from A. Gregorczyk,
A. Mostowski, C. Ryall-Nardzewski, "Definability of sets of models of
axiomatic theories", Bulletin of the Polish Academy of Sciences (series
Mathematics, Astronomy, Physics), vol. 9(1961), pp. 163–7 that the
resulting space is Polish. There is a continuous action of the infinite
symmetric group (the collection of all permutations of the natural
numbers with the topology of point wise convergence) which gives rise to
the equivalence relation of isomorphism. Given a complete first order
theory T, the set of structures satisfying T is a minimal, closed
invariant set, and hence Polish in its own right."
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