Skolems Paradox shows an ostensible conflict between Cantor's Thoerem (CT) and the downward Löwenheim–Skolem Theorem (ST).
CT: for any set $A$, the powerset of $A$, $P(A)$, has a strictly greater cardinality than $A$. Cardinality is in terms of bijections: two sets have the same cardinality iff there exists a bijection between them. A set $A$ is countable iff there exists a objection between $A$ and the set of naturals $\omega$. Since (by CT) no function surjects $\omega$ onto its powerset, we learn that $P(\omega)$ is uncountable. Thus CT generally tells us some sets are uncountable.
ST: If a countable first-order theory has an infinite model, it has a countable model. The standard axiomatization of set theory, ZFC, is such a theory.
Assume ZFC has a model (which must be infinite).
By ST, ZFC has a countable model 𝔐.
By CT, we can deduce the existence of uncountable sets from ZFC.
Therefore, there must be some set A ∈ 𝔐 such that 𝔐 satisfies $A$ is uncountable. That is, there is no bijection $f \in$ 𝔐 between $A$ and $\omega$.
However, since 𝔐 has a countable domain, there are only countably many elements available to be members of A.
Thus A appears both countable and uncountable.
Some good links I found:
1. How did first-order logic come to be the dominant formal logic? (and comments)