Date: Jan 29, 2013 6:32 PM Author: Zaljohar@gmail.com Subject: A philosophical interlude into sets and mathematics A set is a singular entity that corresponds to a totality of singular

entities or

otherwise signifies their absence.

So set-hood is about framing plurality of singular entities within a

singular frame.

A set may be understood as a 'name' for some plurality of singular

entities. But

this would lead to immense naming under standard setting of ZFC, since

this

would obviously lead to "infinite" naming procedures, and even

'uncountably' so

which is something not easy to grasp.

Another way of understanding sets is in a more 'active' manner, so

unlike the

above rather 'passive' context, we rather think of sets as

'collectors' and the

pluralities they collect as 'collections' of course of singular

entities. Clearly

a collector is a 'singular' entity, so this confers with the above

rather abstract

definition of sets.

Of course the 'collector' setting doesn't naturally explain

Extensionality. An

aggressive fix would be to understand sets as "Essential" collectors,

which

postulates the existence of a 'collector' for each collectible

plurality such that

EVERY collector of the same plurality must 'involve' it to be able to

collect that

plurality. Under such reasoning it would be natural to assume

uniqueness of

those kinds of collectors. So ZFC for example would be understood to

be about

collectors of infinite collections, and not about immense kind of

descriptive

procedural discipline.

So for example under the essential collector explanation of sets, one

would say

that ZFC claims the existence of 'uncountably' many essential

collectors of

pluralities (i.e. collections) of natural numbers, most of which

already exceeds

our 'finite' human vocabulary, and even exceeds a 'countably' infinite

vocabulary.

The collector interpretation despite being somehow far fetched still

can explain

a lot of what is going in set/class theories in a flawless spontaneous

manner. It

is very easy to interpret for example non well foundedness, also it is

easy to

understand non extensional versions, even the set theoretic paradoxes

are

almost naively interpreted.

Of course as far as interpret-ability of what sets stands for this is

not a fixed

issue, one is left free to choose any suitable interpretation that

enable him to

best understand what's going on in various set/class theories and

different

manipulations and scenarios involved.

That was a philosophical interlude into what sets are. On can

postulate the

'possibility' of Ontologically extending our physical world with the

world of sets

and even consider all possibilities of so extending it. A minimal

requirement for

such 'possibility' setting is of course to have a 'consistent' record

about such

assumptionally existing entities, since inconsistent record virtually

rule out

having such possibilities.

Would mathematics be the arena of such Ontological extension? How

would that

relate to increasing our interaction with the known physical world?

Zuhair