And I haven't gotten to the trichotomy axiom quite yet, and how it applies to binary relations. So I'm assuming in order for the membership relation to be considered as an order relation it must be trichotomous, which makes sense. Very interesting.
Aha! Thank you so much. I was spending too much time studying ordinals, in which membership is a linear ordering before thinking of this question.
Sometimes thinking too deeply about another topic can lead me into a quagmire such as these.
In the first example, I am not implying in any way that a transitive set cannot be infinite, it is the way in which it is infinite that seems counter-intuitive.
Let B be a subset of A having no least membered element.
Then if b is an element of B, then there exists c in B such that c is...
Example of transitive but not well ordered set needed!
My question pertains to the definition of ordinals. According to Thomas Jech's edition of set theory, a set is ordinal if it is both transitive and well ordered by membership. I've been poking around trying to find an example of a set which...
Alright, it seems logical that in ZFC set theory, a set cannot be a member of itself. My question pertains to the definition of ordinals. According to Thomas Jech's edition of set theory, a set is ordinal if it is both transitive and well ordered by membership. I've been poking around trying...