The usual axioms of $\mathbb{R}$ consist of the ordered field axioms plus the least upperbound property. Because the least upperbound property is a statement that quantifies over sets of real numbers, people often say that it is a second-order logic statement, which makes the theory of real numbers categorical.
However, from the point of view of ZFC set theory, sets are simply objects of set theory, so quantification over sets (like in the least upperbound property) is a first-order statement. Then the theory of real numbers cannot be categorical.
The same kind of confusion can be posed when we're talking about natural numbers. The second-order logic Peano axioms include the induction axiom, which quantifies over properties of natural numbers. But we can recast this as a statement that quantifies over sets in the framework of set theory in first-order logic, and then it isn't clear whether or not (a certain version of) the Peano axioms leads to a categorical theory.
I would want someone with more understanding than me to clarify this situation. What is the right, straightforward way of thinking about this that steers clear of confusion?
Best Answer
In a given model of $\sf ZF$, the real numbers are unique up to isomorphism. But as soon as you reduced the problem to the first order theory of $\sf ZF$, you essentially have to contend with the fact that different models of $\sf ZF$ can exist, and they may have wildly different real numbers.
In some models the reals will be countable, maybe because the model itself is countable. In another model there will be many more reals, perhaps because we took an ultrapower and blew the (external size) of the reals of the model to be much larger than the continuum is supposed to be. And maybe some model contains all the reals.
All of these options are possible and plausible. But each model thinks that its real numbers are the unique Dedekind complete field. Even if we know the truth. The same goes for the second order Peano axioms.