Are you saying that there are no axioms in set theory? You said:

***There are no axioms whatever in Dick's theory, additional or otherwise***

And then you implied that his theory is based on set theory.
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The shift symmetry is the assertion that his theory is independent of where he defines zero to be in his coordinate system.

I believe that this assertion, (I call it an assumption, but Dick claims that there are no assumptions in his theory) is equivalent to covariance in physics and that is the reason that he can derive equations that have been previously derived by physicists using other assumptions.
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I never heard of mathematics without axioms. Are you saying that you can define fields in set theory that have no relationship to each other. Now that is something I would like to see in a journal.

Regards,

Richard