# Does Löwenheim-Skolem require Foundation in any way?

As title states, I'm curious whether Löwenheim-Skolem (in either of its upward or downward versions) necessitates some implicit use of Foundation. The usual presentation makes quite clear the reliance of LS on $$\mathsf{AC}$$ / Zorn's Lemma and as far as I've understood, LS itself is basically equivalent to $$\mathsf{DC}$$. Is there any reliance on Foundation throughout the proofs?

• No, no foundation. They’re not equivalent to DC though. I think some weak version of DLST is, but the strongest phrasings of both are equivalent to full AC, if I recall correctly. Commented Jul 17 at 23:20
• The statement that every infinite set admits a group is equivalent to AC: mathoverflow.net/questions/12973/… Commented Jul 17 at 23:30
• @Lucenaposition It depends exactly how you phrase LS. For example, "Every satisfiable first-order theory with an infinite model has arbitrarily large (= admitting an injection from a given set) models" doesn't give every set a group structure. In fact, ZF alone proves that every set injects into a group (just take the free group on the given set). Commented Jul 18 at 0:41
• My formulation is "For every satisfiable first-order theory $T$ with an infinite model and every infinite set $S$, it is possible to make $S$ a model of $T$ (by adding the relations and predicates." Commented Jul 18 at 0:45

No, there is no use of the Axiom of Foundation in standard proofs of the Löwenheim-Skolem theorem.

To see this, you could of course just read a proof and check for yourself.

But here's another way to see that the Löwenheim-Skolem theorem (like almost all theorems of mathematics outside of set theory) could not possibly depend on Foundation.

$$\mathsf{ZFC-Foundation}$$ proves that every set is in bijection with a well-founded set (indeed, with an ordinal). Hence (for any language $$L$$), any $$L$$-structure is isomorphic to one whose domain is well-founded. Thus, if an $$L$$-theory $$T$$ has infinite models, then it has infinite models in $$\mathrm{WF}$$, the class of well-founded sets. Thus, by Löwenheim-Skolem relativized to $$\mathrm{WF}$$, it has models of arbitrary infinite cardinalities.

This kind of meta-argument shows that any theorem which is provable in $$\mathsf{ZFC}$$ and is "isomorphism invariant" is in fact provable in $$\mathsf{ZFC-Foundation}$$.