The following quote is from the book “What is Mathematics Really?” by Reuben Hersh
“0 (zero) is particularly nice. It is the class of sets equivalent to the set of all objects unequal to themselves! No object is unequal to itself, so 0 is the class of all empty sets. But all empty sets have the same members….none! So they’re not merely equivalent to each other…they are all the same set. There’s only one empty set! (A set is characterized by its membership list. There’s no way to tell one empty membership list from another. Therefore all empty sets are the same thing!) Once I have the empty sets, I can use a trick of Von Neumann as an alternative way to construct the number 1. Consider the class of all empty sets. This class has exactly one member: the unique empty set. It’s a singleton. ‘Out of nothing’ I have made a singleton set…a “canonical representative” for the cardinal number 1. 1 is the class of all singletons…all sets but with a single element. To avoid circularity: 1 is the class of all sets equivalent to the set whose only element is the empty set. Continuing, you get pairs, triplets, and so on. Von Neumann recursively constructs the whole set of natural numbers out of sets of nothing. ….The idea of set…any collection of distinct objects…was so simple and fundamental; it looked like a brick out of which all mathematics could be constructed. Even arithmetic could be downgraded (or upgraded) from primary to secondary rank, for the natural numbers could be constructed, as we have just seen, from nothing…ie., the empty set…by operations of set theory.” Any comments or opinions on whether this theory is the basis for the natural numbers and their relations as is described in the quote above? Thanks -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to email@example.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.