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Topic: A philosophical interlude into sets and mathematics
Replies: 2   Last Post: Feb 14, 2013 12:56 PM

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Posts: 1,635
Registered: 2/27/06
Re: A philosophical interlude into sets and mathematics
Posted: Feb 14, 2013 12:56 PM
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On Jan 29, 6:32 pm, Zuhair <> wrote:
> 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

How do you define philosophy and mathematics, and how can you have
both? If you want mathematics, you have to have formal primitives and
ways to combine them to create something new. Vague synonyms may be
ok for philosophy, but are not mathematics, are they?


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