On mereological counterparts of some principle for sets

Rafal Gruszczynski

Abstract


In this paper we deal with the following problem: what are consequences of adopting the following property of Cantorian sets:

∀z(z is element of the set of all S-ses → z is an S)

for mereological sets?

We show that the abovementioned principle does not hold for any notion of mereological set considered in literature. Further we prove that in case of some classical definitions of mereological set, enriching the theory of such sets with its counterpart leads to trivial, one-element theories. We also consider some less popular (but more natural) definition of mereological set in the form of the so called aggregate of objects and prove that in case of this adoption of the counterpart of the Cantorian principle reduces proper part of relation (i.e. one which is irreflexive) to set theoretical ∈.

In the introduction we present an informal argument for some unwelcome consequences of adopting the principle for mereological sets as defined by means of mereological sums. In the sequel we are more formal and we turn to application of tools of mathematical logic to analyze the problem in its full scope.


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