Re: Singleton as arbitrary

[Dusko Pavlovic <dusko@kestrel.edu>]

Colin McLarty wrote:

> >It comes about as a part of the initial algebra structure
> >on Godel's cumulative hierarchy, just like the successor comes about in NNO.
>
>         Well, something *like* Peano's operation occurs in that initial algebra
> structure, but that is not much to the point.

[snip]

>         It is a recent idea that given any set x there is some set {x}. Bill
> traces it to Peano. It plays no role in ordinary mathematical practice, and
> is unnecessary in set theory. It does not exist in categorical set theory.

but didn't joyal and moerdijk actually write a book about it? i think they call it
successor, but the standard model is x|-->{x}. (or did i mix it all up?)

> >Also, I somehow came to think of set theory as *tree representations of
> >abstract sets*, much like vector spaces are used for group
> >representations.
>
>         The whole point of group representations is that each group has many of
> them. The classical Lie groups are given as groups of linear
> transformations in the first place. The power of representation theory is
> to relate these with *other* representations of the same groups.
>
>         Each ZF set has exactly one membership tree. Thus the "representation"
> cannot do anything like what group representations do.

i didn't say that set theory provides tree representations of ZF sets; ZF sets
*are* trees (or acyclic rooted graphs). i said that set theory provides tree
representation of *abstract* sets. think of lazy natural numbers, flat natural
numbers, finite chains, all of them different *and useful* representations of the
same abstract set.

> And obviously it
> plays no role in ordinary math practice.

the words "obviously" and "practice" don't go together well. 20 years ago, it
seemed obvious that complexity theory was mostly an academic whim. nowadays, the
security infrastructure built upon it is a critical part of the engineering
practices, and the very life of the net. large cardinals may still find unexpected
applications, say in establishing the new tax policies =;0

all the best,
-- dusko