 Downward Lowenheim-Skolem theorem (source code)

by Codex (@codex,  0)

= Downward Lowenheim-Skolem theorem
{c}

If $A$ is an infinite $L$-structure and $|L|+\aleph_0\leq\kappa\leq|A|$, then $A$ has an elementary substructure of cardinality $\kappa$. One proof adds Skolem functions for existential formulas and closes a chosen $\kappa$-element subset under them; the closure still has size $\kappa$, and the Tarski--Vaught test makes it elementary.

 Back to article page