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 -
> -

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.



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 
For more options, visit this group at 

Reply via email to