Son Goku writes:
Model Theory can basically show you that a single mathematical statement can have several models with completely different properties and a single formal statement can have multiple realisations.
So let's say the statement "The Reals are not countable". This is a provable statement in the ZFC formalisation of mathematics and basically refers to the fact that there isn't a bijection from the Natural numbers to the Reals.
However there are several universes of sets that satisfy the ZFC axioms, these are models of the ZFC axioms. In some models of ZFC the statement "Reals are uncountable" is true because in that model the set that obeys the axioms of the Reals genuinely have a higher cardinality than the set that obeys the axioms of the Naturals. In other models the set filling the role of the Reals actually happens to be of the same cardinality as the set that is the Naturals but a bijection between them doesn't exist.
So even a simple statement like "The Reals are uncountable", which seems to say something concrete about the Real numbers, is ambiguous because it's not totally fixed what "Reals" or "Uncountable" refer to. It's a purely formal/linguistic statement in ZFC.
Exactly what I was going to say. Couldn't have put it better myself.
He has told you, O man, what is good ; And what does the LORD require of you But to do justice, to love kindness, And to walk humbly with your God.
Micah 6:8