 To get us started on the idea that foundationalist metaphysicians could accept violations of well-foundedness, consider an important observation by Bliss (2013: 416), which also comes up, e.g., in Rabin &amp; Rabern (2016: 362). She distinguishes between finite and well-founded grounding chains where a finite grounding chain not only terminates in something fundamental, but is also such that we can reach the fundamental entities in a finite number of steps from anywhere in the chain. According to this approach, a well-founded grounding chain is indeed one that is grounded in something fundamental, but it may itself be infinitely long. So, on this terminology, finite grounding chains are always well-founded, but a well-founded grounding chain could be infinite. But note that this already violates set-theoretic well-foundedness as defined above. To demonstrate this, consider an infinite chain of dependence \(f &lt; \ldots d_{3} &lt; d_{2} &lt; d_{1}\), where the chain of dependent entities \(d_{n}\) terminates in some minimal element f. Now, if we take a subset of that chain of dependence without the minimal element f, then we are left with a chain that lacks a &lt;-minimal element, hence violating the set-theoretic definition of well-foundedness.