A total order of some set such that every nonempty subset contains a least element.: The structure (∅, ∅) is a well-order. #* 2014, Abhijit Dasgupta, Set Theory: With an Introduction to Real Point Sets, Springer (Birkhäuser), [https://books.google.com.au/books?id=u06-BAAAQBAJ&pg=PA378&dq=%22well-order%22%7C%22well-orders%22&hl=en&sa=X&ved=0ahUKEwjT077U5pPcAhVIybwKHUwTCnsQ6AEIPjAE#v=onepage&q=%22well-order%22%7C%22well-orders%22&f=false page 378],
#*: Definition 1226 (Von Neumann Well-Orders). A well-order <math>X</math> is said to be a von Neumann well-order if for every <math>x\in X</math>, we have <math>x=\{y\in X\vert y< x\}</math> (that is <math>x</math> is equal to the set <math>\mathrm{Pred}(x)</math> consisting of its predecessors). #*: Clearly the examples listed by von Neumann above, namely
#*:: <math>\empty,\quad \{\empty\},\quad \{\empty, \{\empty\}\},\quad \{\empty, \{\empty\}, \{\empty, \{\empty\}\}\},\quad\dots</math>
#*: are all von Neumann well-orders if ordered by the membership relation "<math>\in</math>," and the process can be iterated through the transfinite. Our immediate goal is to show that these and only these are the von Neumann well-orders, with exactly one von Neumann well-order for each ordinal (order type of a well-order).