Prop. 1 of Proclus' Elements of Theology and Brouwer's intuitionism

The proof of the first proposition of Proclus' Elements of theology is among the most difficult to understand from a formal point of view.  Here is out attempt to make some sense of it using concepts which are also employed in Brouwer's intuitionism (or certain forms of finitism) -  the proof then assumes a structure somewhat like the standard proof of König's lemma.

The proposition reads: every multitude partakes in some way or another of the One.  We take 'multitude' to be represented by a mereological relational system in the form of a tree.

Consider the following interpretation. Proclus assumes that no tree  can have more than a countably infinite number of branches because, for him,  there is no infinity greater than countable infinity (the cardinality of the natural numbers). 

Hence there does not exist a tree in which every node has at least one successor and a fortiori in which each node has infinitely many successors- because then the set of branches would be of the cardinality of the continuum.

Proclus' proposition attempts to characterize trees with countable many branches.

Here are at least three types. Type 1 may have finitely many infinitely branching nodes but all branches of finite length.  Thus it participates of unity in a type 1 way (we may think of the terminal node as a 'unity').

Type 2 may have infinite branches but only finitely many nodes with more than one branch passing through them. Would Proclus accept this ? What are we to make of such chains (perhaps they express return-to-self) ? 

Type 3 has finitely branching nodes and finite length branches (what in intuitionism is called a 'barred spread').   Note that in a finitely branching tree if the length of the finite branches is not bounded then - in classical mathematics - König's lemma implies that the tree has an infinite branch.  The contrapositive of this lemma - called the fan theorem - is in fact intuitionistically valid. Thus Type 3 trees must  have bounds for the length of their branches. This is certainly a participating in 'unity' and 'limit' !