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A philosophical interlude into sets and mathematics
Posted:
Jan 29, 2013 6:32 PM


A set is a singular entity that corresponds to a totality of singular entities or
otherwise signifies their absence.
So sethood 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 interpretability 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



