Definition talk:Strictly Well-Founded Relation

The symbology defines a minimal element not a least element.

The difference is that:
 * for a minimal element (they may be plural), no element is smaller than it
 * for a smallest element, it is smaller than all other elements.

The two necessarily coincide only when the relation in question is a total ordering (and there's result out there that proves it).

I believe the words here are correct but the symbology is wrong - you may want to review it. I have a reputation for always being wrong so please take everything I say here with suspicion. --prime mover 08:14, 11 August 2012 (UTC)


 * Good catch. The symbols were correct, but the description was wrong. --Andrew Salmon 19:40, 11 August 2012 (UTC)

Am I correct that a foundational relation is antireflexive?

Also, it seems that the empty relation is foundational, which seems a bit strange to me... --Lord_Farin (talk) 21:23, 20 August 2012 (UTC)


 * I'm not sure but I think this is just reiterating Definition:Well-Founded. I believe (I may be wrong) that it can be proved that if $\preceq$ is foundational (I'm deliberately using the "ordering" rather than "strict ordering" as that's what most sane authors use) then $\left({S, \preceq}\right)$ is well-ordered and therefore totally ordered, and so "minimal" and "smallest" then mean the same thing. So apart from the fact that this definition applies to classes and the existing page is defined solely in the context of sets, there is actually no difference between these definitions at all. But, as I say, there may be a subtlety here in which I am proven wrong. --prime mover (talk) 21:28, 20 August 2012 (UTC)


 * That's what I'd hope, but there are no impositions on transitivity, hence my caution. (I haven't been able to find an example of a found.rel. not being a poset, but I'm still searching.) --Lord_Farin (talk) 21:35, 20 August 2012 (UTC)


 * Under the current def, $\{a \prec b, b \prec c\}$ on $\{a,b,c\}$ is a found. rel. (any subset contains elements which are not on the RHS of either inequality). --Lord_Farin (talk) 21:38, 20 August 2012 (UTC)


 * I would say that minimal elements should not be restricted to posets. A minimal element (for strict orderings) $y$ of $B$ only need satisfy $\forall x \in B: \neg x \prec y$.  That's at least the sense in which I have been using it.  Minimal elements are very closely related to foundational relations like $\in$, which are, in general, not transitive. --Andrew Salmon (talk) 00:37, 21 August 2012 (UTC)


 * "Very closely related" is imprecise. Either the concept of a minimal element is applicable to a non-transitive relation, in which case the existing page should be amended to make room for this under an "also defined as" section, or it's not, in which case the "foundational relation" definition should be amended so as not to bring in the concept of a "minimal element". --prime mover (talk) 05:23, 21 August 2012 (UTC)


 * Sorry. I meant, "is" applicable, absolutely. --Andrew Salmon (talk) 05:33, 21 August 2012 (UTC)


 * I have added a note to the page Definition:Minimal Element as needed, then. --prime mover (talk) 06:13, 21 August 2012 (UTC)

Antireflexive and asymmetric
From the definition, it follows directly that $\prec$ has got to be antireflexive and asymmetric, while (as we have also seen) it also does not have to be transitive. In fact, as defined here, the Definition:Null Relation is foundational. Have I got it? --prime mover (talk) 06:23, 21 August 2012 (UTC)


 * At least it equals what I deduced, if that counts for something. --Lord_Farin (talk) 06:24, 21 August 2012 (UTC)


 * Yes, this is true. --Andrew Salmon (talk) 00:33, 22 August 2012 (UTC)


 * Both statements (antireflexive and asymmetric) are consequences of Foundational Relation has no Relational Loops. --Andrew Salmon (talk) 06:41, 22 August 2012 (UTC)

More set/class inconsistency
Relational structure is currently only defined for sets. Dfeuer (talk) 21:30, 24 December 2012 (UTC)

Fixing/expanding
This definition works fine for defining a foundational relation in ZF (in classical logic). However, it does not work as the definition of a foundational relation on a class unless the axiom of foundation is accepted. Otherwise we need what Smullyan &amp; Fitting happen to call a "well-founded relation" (just an alternative term for the same thing) and define as a relation on a class such that every subclass (not just subset) has a minimal element. With AoF, these are equivalent, because AoF implies that every set has a rank, and it's possible to build from there to the more precise definition.

It also does not work for intuitionistic logic, where a still-stronger formulation is required to be able to do induction. Specifically, the usual way to do induction classically is to start with the assumption that the theorem is false, conclude that there must be a minimal element at which it fails, and derive a contradiction. But this approach inherently requires LEM. The intuitionistic definition, therefore, is that a well-founded/foundational relation is one over which well-founded induction works. I'm no intuitionist, but I figured I should mention it anyway. --Dfeuer (talk) 15:58, 5 April 2013 (UTC)