Re: Why binary products are ordered

[Dusko Pavlovic <dusko@kestrel.edu>]

Hi.

I am also trying to catch up, perhaps belatedly, with the "spurious
ingredients" thread, but I am quite lost in some parts.

> But woven into Charles' argument is what Bill has called the "totally
> arbitrary singleton operation of Peano."

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.
Aren't all our inductive constructions based on such operations, including the
software we are using to run this conversation?

I would really appreciate help with this.

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. And more than that, the
essence of our thinking: Isn't every metaphor, as a deviation from the abstract
view, built of spurious elements? Isn't every novel a bunch of lies, things
that never happened, put together to tell some truth? Can we really define
cartesian product without the spurious elements?

I am sure we can, but it would be good to know more precisely how to
distinguish the spurious from the authentic elements. Otherwise, we may end up
"slinging back and forth ill-defined epithets", like i am probably doing now.

With apologies, and best wishes,
-- Dusko

> Surely anyone insisting on names like 1 and 2 or red and blue for the
> projections of binary product is backsliding into the ZFvN tarpit of
> spurious rigidified membership.  If this backsliding really is inevitable
> as Charles seems to be saying, how does one reconcile this with Bill's
> view of "rigidified membership" as "mathematically spurious"?
>
> Must mathematics accept the spurious, in this or any other case?
>
> Vaughan