This is a followup question to Does foundation/regularity have any categorical/structural consequences, in ZF?

As shown in answers to that question, the axiom of foundation (AF, aka regularity) has consequences over ZF–AF that can be formulated purely in structural terms (i.e. using sets only in ways that respect isomorphism).

In particular, it implies the statement “Every set can be embedded into a set equipped with a well-founded extensional relation”, which Mike Shulman has called the *axiom of well-founded materialisation*; and (by the arguments in that question) this axiom should have the same structural consequences as foundation.

This statement is a very mathematically reasonable one. However, I don’t think I have ever seen this or anything similar appearing in practice in ordinary mathematics — only in logical investigations like this one, where it occurs specifically because of its relation to AF. So my question here is:
**Does the axiom of foundation have any consequences (over ZF–AF) that have appeared in ordinary mathematics?**

Slightly more precisely:

By “consequence of foundation over ZF–AF”, I of course mean principally a statement that’s provable in ZF, but not in ZF–AF. But I’d also be interested in e.g. a proof in ZF that doesn’t go through in ZF–AF and can’t be trivially modified to do so.

By “ordinary mathematics”, I mean… this of course is subjective, and mostly I think “we know it when we see it”, and hope that most of us will find our judgement on it mostly agrees. But to articulate my own intent here a little: a sufficient (but not necessary) condition to would be that the statement has been considered in a field of mathematics other than logic; necessary conditions would be that it’s purely structural, and that it’s been considered outside work specifically investigating foundation (or related axioms) or modelling set theories that include foundation. To illustrate the non-necessity of the “outside logic” criterion — I would certainly be happy to consider consequences in e.g. general classical model theory.

See e.g. Where in ordinary math do we need unbounded separation and replacement? for a similar question, about different axioms, and relevant discussion of what should be considered “ordinary mathematics”.

7more comments