Skolem's paradox reveals that 'truth in all models' is model-relative: a sentence about uncountability can be true in the intended model yet satisfied in a countable model via different satisfaction relations.
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The apparent paradox that arises because a countable model of ZFC can internally satisfy the theorem that uncountable sets exist, despite the model's domain being countable from an external perspective
Uncountability(as the property being discussed in the sentence)
A quality of infinite sets that are so large they can't be listed or matched up with regular counting numbers, no matter how hard you try (like the infinity of all real numbers).
intended model(Model theory / philosophy of mathematics)
The canonical structure that a theory is meant to describe; for ZFC, this is the proper class V with the ∈ relation
model-relative(describing properties in logic and mathematics)
Dependent on which particular version or interpretation of a system you're looking at; something can be true in one model but not in another.