> Isn't that the case for pretty much everything that you can twist them in any direction by the choice of axioms?
By models in this context are usually meant models of this one particular set of axioms you’re talking about at the moment (considered in some sort of ambient metatheory that isn’t always spelled out explicitly but you can go in that direction too if you want). For example, you can construct a model of PA in ZF that is morally { {}, {{}}, {{{}}}, ... } for the elements (that was not a syntactically valid formula in ZF, but you can prove a critter that is intuitively described by it exists nevertheless) and x \mapsto {x} for the successor, even though PA says nothing about sets, functions, or anything of that sort. Basically, for “model” read “implementation” (an “interpretation” of a thing within another thing is a synonymous term that actual logicians occasionally use).
(And yes, there are models of PA inside ZF that are substantially different from that one[1].)
> Isn't that the case for pretty much everything that you can twist them in any direction by the choice of axioms?
By models in this context are usually meant models of this one particular set of axioms you’re talking about at the moment (considered in some sort of ambient metatheory that isn’t always spelled out explicitly but you can go in that direction too if you want). For example, you can construct a model of PA in ZF that is morally { {}, {{}}, {{{}}}, ... } for the elements (that was not a syntactically valid formula in ZF, but you can prove a critter that is intuitively described by it exists nevertheless) and x \mapsto {x} for the successor, even though PA says nothing about sets, functions, or anything of that sort. Basically, for “model” read “implementation” (an “interpretation” of a thing within another thing is a synonymous term that actual logicians occasionally use).
(And yes, there are models of PA inside ZF that are substantially different from that one[1].)
[1] https://golem.ph.utexas.edu/category/2019/06/nonstandard_mod...