That was imprecise, but I was trying to comment on this part of the dialogue using the language that it had established:

Argh! No, damn it, I live in the set theory that really does have all the subsets, with no mysteriously missing subsets or mysterious extra numbers, or denumerable collections of all possible reals that could like totally map onto the integers if the mapping that did it hadn't gone missing in the Australian outback -

I was also commenting on this part:

Screw set theory. I live in the physical universe where when you run a Turing machine, and keep watching forever in the physical universe, you never experience a time where that Turing machine outputs a proof of the inconsistency of Peano Arithmetic.

The point I was trying to make, and maybe I did not use sensible words to make it, is that This Guy (I don't know what his name is - who writes a dialogue with unnamed participants, by the way?) doesn't actually know that, for two reasons: first, Peano arithmetic might actually be inconsistent, and second, even if it were consistent, there might be some mysterious force preventing us from discovering this fact.

I just don't understand yet what you mean by living in a model in the sense of logic and model theory, because a model is a static thing.

Models being static is a matter of interpretation. It is easy to write down a first-order theory of discrete dynamical systems (sets equipped with an endomap, interpreted as a successor map which describes the state of a dynamical system at time t + 1 given its state at time t). If time is discretized, our own universe could be such a thing, and even if it isn't, cellular automata are such things. Are these "static" or "dynamic"?