Date: Jan 29, 2013 9:52 PM Author: William Elliot Subject: Re: A philosophical interlude into sets and mathematics On Tue, 29 Jan 2013, Zuhair wrote:

To curtain the ad infinitum:

"Get ready. Get set. Go!"

> 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?