I think so. I’m confused as to whether these results can apply to the actual standard natural numbers, or just partly-weird sets which some model of set theory believes are the standard numbers—but I think it’s the second one. The paper itself says that “satisfaction is absolute,”
whenever the formula φ has standard-finite length in the meta-theory (which is probably closer to what was actually meant by those asserting it)
which appears to mean that none of the statements being disagreed about are actual mathematical statements, but rather “weird numbers” that the models think encode statements. (Note that any other way of defining mathematical formulas would probably be at least as ambiguous as the natural numbers.)
My understanding passes the first test which occurs to me, namely, it would not allow a model of set theory to realize that it has the wrong natural numbers. (If my interpretation predicted that a model with weird numbers could look at another model which disagrees about arithmetical truth and realize, ‘Wait, these disagreements only arise when we allow weird numbers, so my numbers must be partly weird,’ then I would have to be wrong.) Evidently if even one of two models can look at the satisfaction relation that the other uses to define arithmetical truth (for example), then they must agree.
I think so. I’m confused as to whether these results can apply to the actual standard natural numbers, or just partly-weird sets which some model of set theory believes are the standard numbers—but I think it’s the second one. The paper itself says that “satisfaction is absolute,”
which appears to mean that none of the statements being disagreed about are actual mathematical statements, but rather “weird numbers” that the models think encode statements. (Note that any other way of defining mathematical formulas would probably be at least as ambiguous as the natural numbers.)
My understanding passes the first test which occurs to me, namely, it would not allow a model of set theory to realize that it has the wrong natural numbers. (If my interpretation predicted that a model with weird numbers could look at another model which disagrees about arithmetical truth and realize, ‘Wait, these disagreements only arise when we allow weird numbers, so my numbers must be partly weird,’ then I would have to be wrong.) Evidently if even one of two models can look at the satisfaction relation that the other uses to define arithmetical truth (for example), then they must agree.