Proclus' Elements Prop. 1

A nice edition of Proclus' Elements of Theology.

There would seem to be a problem with Proclus' proof from the point of view of modern set theory. If we take the natural numbers N then the even numbers E are a 'part' of N and yet have the same cardinality as N. So in this case in what sense is the whole N 'greater' than the part E ?

We could  do an idealist and phenomenological reading of this proposition, specially in light of Husserl's Philosophy of Arithmetic which is actually a treatise on some of the highest categories and operations of the understanding, in particular the act of combination or synthesis. So let us take Proclus' plêthos to be any object of consciousness which can be called an aggregate.

We begin with the psychological characterization of that abstraction which leads to the (authentic) concept of the multiplicity, and subsequently to the number concepts. We have already indicated the concreta on which the abstracting activity is based. They are totalities of determinate objects. We now add: "completely arbitrary" objects. For the formation of concrete totalities there actually are no restrictions at all with respect to the particular contents to be embraced. Any imaginable object, whether physical or psychical, abstract or concrete, whether given through sensation or phantasy, can be united with any and arbitrarily many others to form a totality, and accordingly can also be counted. For example, certain trees, the Sun, the Moon, Earth and Mars; or a feeling, an angel, the Moon, and Italy, etc. In these examples we can always speak of a totality, a multiplicity, and of a determinate number. The nature of the particular contents therefore makes no difference at all. (Husserl, PA, p. 17 Dillard tr.)

Thus we must distinguish between concrete multiplicities and abstract multiplicities.  Husserl explores the the aspects of combination and synthesis involved, including syntheses of syntheses and so forth. Proclus' conclusion would then seem to be follow from the limitations of such syntheses operations for human consciousness. The problem is that this is relative to the human mind only.

However perhaps we can glean some insights into neoplatonic epistemology and psychology from the interesting work by Gregory MacIsaac, The Soul and Discursive Reason in the Philosophy of Proclus, 2001.

Proclus' proof (his assumption that there is nothing greater than the apeiron) seems to suggest the axiom of foundation of ZFC.  If we view a set as a tree ordered by the membership relation then all paths starting at the root (the original set) must eventually end.