> > Aatu Koskensilta kirjoitti: > > > > Because there is a model of set theory which has domain that is > > > > countable and which I think contains all the reals.
That should probably read "and which it thinks contains all the reals", i.e., the model thinks that. No human "I" thinks that.
> > > > > > Well, there isn't. No countable model contains all the reals. > > > > How can the countable model then satisfy all the theorems about the > > reals? > > And especially the theorem "There is uncountably many reals."!
You're entitled to your confusion; don't let teachers talk you out of it too easily. This has been dubbed "Skolem's Paradox". It is not really a paradox but it DOES imply that a first-order model's opinion about questions that are inherently second- order can be, well, inadequate, if not inaccurate.
The root cause of the problem is that the powerset axiom, as applied to infinite sets, is "intended" to require that "all" subsets of the infinite set actually occur in the powerset; but in practice, at first-order, you can satisfy the axioms while only including some (some countable subclass, even) of the subsets in the powerset. or equivalently, only including some of them in the domain of the model. The other subsets (or, in the case of counting the reals, the other bijections) still "exist", platonically, but they are not included in the domain of the model.
Just what kind of first-order axioms you would need to add to ZFC to ensure that the powersets would be fuller and fatter is, unfortunately, still unclear after all these decades.