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?