Precise definition of "well-founded relation"

Question

In Just & Weese's "Discovering Modern Set Theory I: The Basics" (AMS, 1998 edition), we encounter the definition of a well-founded relation (or wellfounded relation) as:

"Let $R$ be a binary relation on a set $X$. We say that $R$ is wellfounded if for every nonempty subset $Y \subseteq X$ there exists a $z \in Y$ such that $\langle{y, z}\rangle \notin R$ for all $y \in Y \setminus \{z\}$. A relation $R$ is strictly wellfounded if it is wellfounded and irreflexive."

This is similar to, but not identical with, the definition as found on Wikipedia, which itself claims to have used Just & Weese as a source: $$(\forall S \subseteq X)\; [S \neq \emptyset \implies (\exists m \in S) (\forall s \in S) \lnot(sRm)]$$

This is pretty much the same, except for the fact that Just & Weese have that awkward "for all $y \in Y \setminus \{z\}$."

The latter definition coincides with how ProofWiki define it on https://proofwiki.org/wiki/Definition:Foundational_Relation, where just to be confusing they call it a "foundational relation" rather than a "wellfounded relation":

That is, $\mathcal R$ is a '''foundational relation on $A$''' iff:

:$\forall s: \left({s \subseteq A \land s \ne \varnothing}\right) \implies \exists y \in s: \forall z \in s: \neg \left({z \mathrel{\mathcal R} y}\right)$

This definition appears to come from Takeuti and Zaring's "Introduction to Axiomatic Set Theory" (1971) but I have not seen it to confirm.

Given that we have that a foundational relation is antireflexive, again from ProofWiki:

https://proofwiki.org/wiki/Foundational_Relation_is_Antireflexive

it appears that the definition as given on ProofWiki is actually for a strictly wellfounded relation as defined in Just & Weese.

What is standard here? Wikipedia seems lax here, and ProofWiki (which I am trying to improve and make rigorous as the object of this exercise) appears incomplete.

So: does anyone know what the current accepted definition of well-founded and strictly well-founded (and similarly foundational and its variants) actually are? I take it for granted that well-founded means the same thing as foundational in this context, but that may be inaccurate, and I need confirmation of that.

I also take on board the fact that the situation complexifies when you expand set theory to class theory, but I have not fought my way through the undergrowth to that particular prize yet.

Answer 1

I looked up currently on my shelf the six textbooks that include the term "well-founded," how they each define well-founded. The general consensus appears to be that the overall purpose of a well-founded relation, however it is defined, is to avoid infinite descending chains. This is even exactly how it is defined in at least two sources:

A binary relation $E$ is said to be well founded iff there are no infinite sequences decreasing with respect to $E$, that is, no infinite sequences $x_0$, $x_1$, $\ldots$ such that $x_1 E x_0$, $x_2Ex_1$, $\ldots$. (Chang & Keisler, Model Theory, 2012 Dover edition)

The model $A$ is said to be well founded if there is no infinite sequence $\langle a_n : n<\omega\rangle$ such that $a_{n+1} \in_A a_n$ for all $n < \omega$. (Simpson, Subsystems of Second Order Arithmetic, 1999)

(Simpson also includes a definition of well founded in relation to trees, namely that a tree is well-founded if it does not contain an infinite path, which is much in the same flavor but in a different enough context that I thought it not exactly applicable.)

The lack of an infinite descending chain is a negative condition, however. Others define it positively, taking the "has a minimal element" route, including Kanamori, Jech, and Takeuti & Zaring (although as you point out, their definition is called a "foundational relation," and they define a well-founded relation to be something slightly different):

For a binary irreflexive relation $R$ with field a set $X$, $R$ is well-founded iff any non-empty $Y \subseteq X$ has an $R$-minimal element. (Kanamori, The Higher Infinite, 2009)

A binary relation $E$ over a set $P$ is well-founded if every nonempty $X \subseteq P$ has an $E$-minimal element, that is $a \in X$ such that there is no $x \in X$ with $xEa$. (Jech, Set Theory, 1978)

(Jech then immediately states as an exercise that it is a theorem of the definition that there are no infinite descending chains.)

Takeuti and Zaring have two definitions of interest here:

Definition 6.21 $$R \text{ Fr } A \overset{\Delta}{\leftrightarrow} (\forall a)\left[a\subseteqq A \wedge a \neq 0 \rightarrow (\exists x \in a) \left[a \cap R^{-1}\{x\}=0\right]\right]$$ $\ldots$ Definition 6.24(1) $$R\text{ Wfr }A \overset{\Delta}{\leftrightarrow} R \text{ Fr }A \wedge (\forall x \in A)\left[\mathscr{M}(A \cap R^{-1}\{x\})\right]$$ (Takeuti & Zaring, Introduction to Axiomatic Set Theory, 1971)

The first is the usual: "$R$ is a foundational relation on $A$ iff each nonempty subset of $A$ has an $R$-minimal element." The second is a little different: they define $\mathscr{M}(A)$ to mean that $A$ is a set (technically, per definition 4.10, $\mathscr{M}(A) \overset{\Delta}{\leftrightarrow} (\exists x)[x=A]$. They allow capital letters to be either individual variables or class symbols.). They write "$R$ is a well founded relation on $A$ iff each nonempty subset of $A$ has an $R$-minimal element and each $R$-initial segment is a set. By an $R$-initial segment of $A$ we mean the class of all elements in $A$ that $R$-precede a given element of $A$ i.e., $A\cap R^{-1}\{x\},\ x\in A$."

Cohen's is probably the most unique:

A set $x$ is well-founded if $x$ belongs to some transitive set $A$ for which a rank function exists. (Cohen, Set Theory and the Continuum Hypothesis, 2008 Dover edition)

(Cohen defines a rank function on a transitive set $A$ is an ordinal-valued function $r$ on $A$ such that for all $x \in A$, $r(x)=\sup\{r(y)\mid y \in x\}$.)

So, it appears as though the general consensus is that a relation is well founded either if it has no infinite descending chains, or if every subset has a minimal element. It looks like you can take your pick depending on what it is you need.