Definition talk:Epsilon Relation

There are a load of things going wrong here. The definition of $E$ is not void, but I second strongly the third point made, i.e. what is the use of $E$ over $\in$. Furthermore, things like Epsilon Relation is Strictly Well-Founded are (at least for the moment) void as a relation (and any relational structure) demands a set to work on, while the $A$ in consideration is a class.

Trying to copy the notation used in Takeuti/Zaring's "Introduction to Axiomatic Set Theory" (the theorem itself is not a theorem in the book, interestingly), I must have created a lot of confusion. The strictly well-founded relation is supposed to be a predicate, taking values of either classes or sets.

The second point is not a problem however (the statement in parentheses), because it is simply void (or at best: yet to be defined how) to talk about $(x,y)\in E$ if $E$ is a class. If $E$ is a set, no problem arises.

As it turns out $\Epsilon$ is a proper class (as its domain is the universe), but that doesn't alter the behavior of $\in$ in this case. The behavior is the same for sets $\in$ sets/proper classes. However, $\Epsilon$ cannot be a member of anything else.

I would say, in general, that the ZF(C) axioms tell us how to use the formal symbol $\epsilon$ (more precisely: the binary function symbol $\epsilon$) of our language in correspondence to everything else. The need for $E$ is at the moment not clear to me. Hopefully, someone can elucidate me on this point. --Lord_Farin 03:36, 26 November 2011 (CST)

Since $\in$ is not a class, we cannot say that it is a foundational relation on any set (what the axiom of regularity is saying for $\Epsilon$), or that it well-orders the ordinals (as I intend to write up soon). We would also not be able to say that \in well-orders the ordinals. Also, in chapter 12 and onward, "Introduction to Axiomatic Set Theory" talks about inner model theory, and $\Epsilon$ is used in several definitions (since, unlike $\in$, you can create a restriction on $\Epsilon$ or intersect it with another set, such as a cross product). --asalmon

Yes, I see how this definition will be useful; the structure will be necessary numerous times in proofs, and thus we'd better name it. Thanks. --Lord_Farin 14:35, 26 November 2011 (CST)