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
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