***The two statements below, which followed each other seem to be contradictory, one saying that Peano Axioms are assumptions, and the other implying that they are math tools and not assumptions.***
The statements are not contradictory. The Peano Axioms are assumptions (or else they are derived from more basic assumptions). The Peano Axioms are also math tools. The mathematical system of arithmetic consists of the assumptions, equivalent to the Peano Axioms, and a set of definitions which produce numbers. This system of arithmetic is a very powerful mathematical tool for solving many kinds of problems.
Of course, if you use the tool, you are accepting the underlying assumptions, and I don't think Dick would deny that.
***Stafford and I have gone round and round about his assumptions- particularly the shift symmetry assumption.***
Yes, I have read that go-around but I have never quite understood what difficulty you see. The way I see it, Dick has accepted the mathematical system of analysis, the rules of logic, and the English language as the tools with which to make and present his derivation. Aside from that, in the derivation itself, there are no further assumptions on his "set of numbers". If one wishes to form an isomorphism between his theorem (sorry Dick) and any perceived universe, then the assumption must be made that the universe must be communicable in order that its features can be expressed as a set of numbers so that the theorem applies to it.
Now, when you start talking about shift symmetry, I have not been able to think of a way of adding anything to what Dick has already told you. If I just come out and say that there is no assumption of shift symmetry made, I don't think you would be convinced. So I won't say it.