On Aug 1, 5:24 am, "Stephen P. King" <stephe...@charter.net> wrote: > On 7/31/2011 7:40 PM, Pzomby wrote: > > > > > 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 > > Hi Pzomby, > > Nice post, but I need to point out that that von Neumann's > construction depends on the ability to bracket the singleton an > arbitrary number of times to generate the pairs, triplets, etc. which > implies that more exists than just the singleton. What is the source of > the bracketing? I have long considered that this bracketing is a > primitive form of 'making distinctions' which is one of the necessary > (but not sufficient) properties of consciousness. > > Onward! > > Stephen- Hide quoted text - > > - Stephen:
The full three paragraphs are from the book. The sentence ‘Once I have the empty sets, I can use a trick of Von Neumann as an alternative way to construct the number 1.’ is Hersh’s words. I was looking for opinions, as you have given, on Hersh’s conclusions. Your comment on ‘making distinctions’ is the direction I was heading in understanding the role of primitive mathematics (sets, numbers) underlying human consciousness. Thanks Pzomby -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.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.
