Re: Singleton as arbitrary

Re: Singleton as arbitrary

[Colin McLarty]

	Dusko Pavlovic points out that various data types representing the natural
numbers by posets are useful. That is true but hardly related to tree
representations in set theory. (For example, these posets are generally not
"extensional" in the sense of membership trees.)                             
	
	When I remarked that the operation x|--> {x} has no role in mathematical
practice and does not exist in categorical set theory, Dusko replied:

>but didn't joyal and moerdijk actually write a book about it?

	Joyal and Moerdijk wrote a book on algebraic characterization of models of
ZF. They use an operation with formal properties like Peano's singleton. So
I have to admit the singleton operation does figure in practice, when the
"practice" is to describe ZF and related set theories. Not otherwise.

	When I said membership trees "obviously play no role in ordinary math
practice" he replied

>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

	Membership trees are hardly the same as the study of large cardinals. The
large cardinals I know of are all described by isomorphism invariant
properties (measurable: an uncountable set k which admits a non-principle
k-complete ultrafilter). So the definitions that ZF set theorists give do
not rely on membership, they are already definitions in categorical set
theory.   

	As to "obvious", we might wish that everything about practice was obscure.
It would free up 'debate' wonderfully. But it is obvious right now that
membership trees in set theory are used only for a handful of technical
theorems in the foundations of set theory. Categorical set theorists also
use them, for equiconsistency results with ZF. 

	I don't claim to *prove* they will never have any other use. Perhaps one
day they will be central to work in PDEs. Perhaps one day (as Philip
Johnson predicts) Bible based biology will produce far greater advances
than materialist science as practiced in recent centuries. I only say such
claims are arbitrary.

best regards, Colin 

Re: Singleton as arbitrary

[Dusko Pavlovic]

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

Singleton as arbitrary

[Colin McLarty]

Dusko Pavlovic <dusko@kestrel.edu> wrote:

>To begin embarassing myself --- I am not sure what Peano's singleton
operation
>is. Is it the map x|-->{x}?
>
>If it is --- why is this operation totally arbitrary, like Bill says? In
>particular, why is it more arbitrary than the successor operation in
arithmetic
>(which Peano used)? 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.

	The point is: successor did not have to wait for NNOs to be defined. It
occurs throughout arithmetic for nearly as long as we have records of
systematic thought. And it is central to all uses of arithmetic today.

	The *singleton subset* idea is also very old: A geometric condition can
define a subset of points in the plane, and perhaps a singleton subset. And
singleton subsets are all over math today for the same reason.

	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.
     

>Also, I somehow came to think of set theory as *tree representations of
>abstract sets*, much like vector spaces are used for group
representations. Is
>this wrong? It seems to me that introducing the external, "spurious" elements
>(eg vectors) is the whole point of 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. And obviously it
plays no role in ordinary math practice.

	I hope no one believes that singletons, or trees, or vectors are
"spurious" per se. Some uses of the ideas are "arbitrary", and some claims
about them are "spurious". 

best, Colin