I apologize in advanced if I'm hopelessly confused...
Skolem's Paradox, I suppose, can be put like this:
$M$ is a countable model of ZFC and $M$ implies the existence of uncountable sets.
I suppose that people find this initially paradoxical because they assume the statement is said from a single, absolute perspective. However, there are (necessarily) two perspectives involved: the inside perspective on $M$ and the outside perspective on $M$. Once these perspectives are separated, we realize that there is no paradox. Consider:
The former conjunct "$M$ is a countable model of ZFC" is necessarily said from an outside perspective on $M$ -- as discussed here. Actually, $M$ can't express its own cardinality at all.
(Is $M$'s inability to express its own cardinality related to $M$'s being a proper class -- namely, that there is no function in $M$ that takes one of $M$'s members onto the universe of $M$?)
Continuing on…Let $N$ be the outside perspective on $M$ such that there is a bijection $f\in N$ between the domain M of $M$ and $\omega^N\in N$.
The latter conjunct "$M$ implies the existence of uncountable sets" is obviously said from $M$'s inside perspective -- after all, $M$ is a model of ZFC and thus must satisfy Cantor's Theorem.
So, we can separate the perspectives of the paradoxical statement above here:
- From $M$'s perspective, $M$ is a proper class and there is some $A\in M$ such that $A$ is uncountable in $M$.
- From $N$'s perspective, $M$ is countable.
These two statements are jointly consistent when we realize that what can be said of a set $B$ is relative to what some model has to say about $B$. And so, the paradoxical statement isn't so paradoxical.
Is there anything wrong with what I have written above? (It took me a long time to learn this stuff, esp. with zero background in set-theory, higher-math, higher-logics, model-theory, etc., so specifically telling me where I am going wrong, if I am, will be a great help and a great relief.)
What I'm really interested in is what can we say about $A$ from $N$'s perspective? Are the following possible:
- $M$ sees $M$ as a proper class, $M$ sees $A$ as uncountable, $N$ sees $M$ as countable, and $N$ sees $A$ as finite.
- $M$ sees $M$ as a proper class, $M$ sees $A$ as uncountable, $N$ sees $M$ as countable, and $N$ sees $A$ as countable.
- $M$ sees $M$ as a proper class, $M$ sees $A$ as uncountable, $N$ sees $M$ as countable, and $N$ sees $A$ as uncountable.
Under what conditions might (1) - (3) be individually possible (obviously they can't be jointly possible)?
I suspect this question might be rather simple. For example, (2) could be possible when $N$ recognizes a bijection both between $M$ and $\omega^N$ and between $A$ and $\omega^N$. (3) could be possible when $N$ recognizes a bijection between $M$ and $\omega^N$ but doesn't recognize a bijection between $A$ and $\omega^N$.
My goal here is to understand/stress the fact that truth in model theory is relative to what particular models have to say about their members. So, I'm trying to see that while $M$ may take $A$ to be uncountable, $N$ can take $A$ to be of any cardinality even under the condition that $N$ sees $M$ as countable.