Transitive closure in Goodman and Quine's system

In the <a href="https://alexanderpruss.blogspot.com/2024/11/goodman-and-quine-and-shared-bits.html" rel="nofollow">previous
post</a>, I showed that <a href="https://sites.pitt.edu/~rbrandom/Courses/2023%20Sellars/Sellars%20texts/Goodman-StepsTowardConstructive-1947.pdf" rel="nofollow">Goodman
and Quine’s</a> counting method fails for objects that have too much
overlap. I think (though the technical parts here are more difficult)
that the same is true for their definition of the ancestral or
transitive closure of a relation.
GQ showed how to define ancestors in terms of offspring. We can try
to extend this definition to the transitive closure of any relation
R over a kind K of entities (say, organisms):

x stands in the transitive
closure of R to y iff x and y are K and x is a part of every object u that has y as a part and that is such that
whenever z is K and is a part of u then every K that has R to z is a part of u.

This works fine if the Ks
do not overlap. But if they do overlap, it can fail. For instance,
suppose we have three atoms a,
b and c, and a relation R that holds between a + b and a + b + c and
between a and a + b. Then any object
u that has a + b + c as a
part has c as a part, and so
(1) would imply that c stands
in the transitive closure of R
to a + b + c, which
is false.
Can we find some other definition of transitive closure using the
same theoretical resources (namely, mereology) that works for
overlapping objects?
No. 
Let’s work in models with a countably infinite number of mereological
atoms and a successor relation between the mereological atoms that
satisfies the standard axioms for a successor relation. (E.g., these
models might correspond to worlds where the atoms pop into existence
sequentially.)

Say that an object y
stands in the one-plus relation to an object x iff x is a part of y and there is exactly one atom of
y that is not a part of x.
An object y is finite
iff there is an atom x such
that y stands in the
transitive closure of one-plus to x.
Given objects x and
y, let Kxy(z)
if and only if z is a fusion
of a part of x and a part of
y.
For nonoverlapping x
and y, let Rxyuv
just in case Kxy(u)
and Kxy(v),
and the overlap of v with
x stands in one-plus to the
overlap of u with x, while the overlap of v with y stands in one-plus to the overlap
of u with y.
Say that x is
equinumerous0 to y provided that x and y don’t overlap and are both finite
and an object consisting of two atoms is in the transitive closure of
Rxy to
the fusion of x and y.
Say that x is
equinumerous to y provided
that they are both infinite or they are both finite and there is a z that is equinumerous0 to each of x and y.

So, we now have equinumerosity defined for collections of atoms.
Given the correspondence between GQ’s system and monadic second-order
logic, if we can define equinumerosity in terms of parthood and the
successor relation, we have violated the result in the answer https://math.stackexchange.com/questions/619131/can-equinumerosity-by-defined-in-monadic-second-order-logic
.
Since the one tool over and beyond parthood and successor relation we
used in the proof is transitive closure, it follows that that could not
have been definable in terms of parthood and the successor relation.

https://alexanderpruss.blogspot.com/2024/11/transitive-closure-in-goodman-and.html