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Re: A philosophical interlude into sets and mathematics
Posted:
Jan 29, 2013 9:52 PM
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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?
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