Can we construct a universal Turing machine (UTM) using only Peano's axioms for addition?
We know that arithmetic without addition is complete, and that adding multiplication makes it incomplete.
I am trying to understand more deeply if there is a link between incompleteness and the existence of a UTM.
It has been shown by Chaitin that a UTM can exists as a Diophantine equation involving multiplication, addition and exponential. So a UTM from full arithmetic is definitely possible. No claims however are made as to if this is the logically simplest or not.
Furthermore, a UTM moves tapes left and right, write a symbol, erase a symbol, or read a symbol. These does not seem to be multiplicative operations.