Date: Jan 29, 2013 6:32 PM
Subject: A philosophical interlude into sets and mathematics
A set is a singular entity that corresponds to a totality of singular
otherwise signifies their absence.
So set-hood is about framing plurality of singular entities within a
A set may be understood as a 'name' for some plurality of singular
this would lead to immense naming under standard setting of ZFC, since
would obviously lead to "infinite" naming procedures, and even
which is something not easy to grasp.
Another way of understanding sets is in a more 'active' manner, so
above rather 'passive' context, we rather think of sets as
'collectors' and the
pluralities they collect as 'collections' of course of singular
a collector is a 'singular' entity, so this confers with the above
definition of sets.
Of course the 'collector' setting doesn't naturally explain
aggressive fix would be to understand sets as "Essential" collectors,
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
plurality. Under such reasoning it would be natural to assume
those kinds of collectors. So ZFC for example would be understood to
collectors of infinite collections, and not about immense kind of
So for example under the essential collector explanation of sets, one
that ZFC claims the existence of 'uncountably' many essential
pluralities (i.e. collections) of natural numbers, most of which
our 'finite' human vocabulary, and even exceeds a 'countably' infinite
The collector interpretation despite being somehow far fetched still
a lot of what is going in set/class theories in a flawless spontaneous
is very easy to interpret for example non well foundedness, also it is
understand non extensional versions, even the set theoretic paradoxes
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
manipulations and scenarios involved.
That was a philosophical interlude into what sets are. On can
'possibility' of Ontologically extending our physical world with the
world of sets
and even consider all possibilities of so extending it. A minimal
such 'possibility' setting is of course to have a 'consistent' record
assumptionally existing entities, since inconsistent record virtually
having such possibilities.
Would mathematics be the arena of such Ontological extension? How
relate to increasing our interaction with the known physical world?