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