Skolem's paradox

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Skolem's paradox is an apparent contradiction that arises from the Lowenheim-Skolem theorem. We can consider a first-order theory containing axioms that appear to not be satisfiable by a countable model, such as union along with "an uncountable set exists". But if there's an uncountable set that satisfies this, then applying the Lowenheim-Skolem theorem downwards, there must be a countable model $M$ of this theory too.

The resolution of this confusion is that there are members $x$ of $M$ which are locally uncountable: that is, $M\vDash``x\textrm{ is uncountable}"$. Since $M$ is in reality countable and transitive, $x$ must be countable, but $M$ thinks that $x$ is uncountable.


An example of this phenomenon is the model $M=\textrm{HF}\cup\{\omega\}$: in this model, there doesn't exist a bijection between $\omega$ and $\mathbb N$, so $M\vDash``\omega\textrm{ is uncountable}"$ even though they are the same set. $\omega=\mathbb N$ is the only infinite set in $M$ and it is not a bijection because every bijection is a relation, i.e. a set of ordered pairs, and ordered pairs are (at most) two-element sets (e.g., $(x, y)=\{x, \{x, y\}\}$ or $(x, y)=\{\{x\}, \{x, y\}\}$ depending on convention) while $\omega$ contains $3$ that has three elements.[ citation needed ]. $\textrm{HF}$ is the class of hereditarily finite sets.

More "natural" examples show up with heights of models, whose ranks are often closed under union, pairing, and ordered pair formation. For example, Arai gives an explicit example of a countable $\sigma$ such that $L_\sigma$ models $\textrm{KP}+``\omega_1\textrm{ exists}\!"$, and also gives a value of $(\omega_1)^{L_\sigma}$. [1]