Relation between sets and their elements

What is the relation between sets and their elements (provided, of
course, they have any)?

Is it the relation of identity or of parthood, or in some sense both of
identity and of parthood?

Are the Fs as the elements of the set of Fs (identical with) the set of
Fs?

The Fs = The elements of the set of Fs = the set of Fs?

And if not, in what sense are the Fs different from and independent of
the set of Fs, whose elements they are?

When I talk of the Fs, do I also necessarily talk of the set of Fs by
doing so?

And if that should be the case, can I nevertheless still talk of the
sets, when there is no such thing as the set of sets?

Can sets lose their elements and keep their identity as distinct
entities, i.e. as entities distinct from the empty set? (I don't think
so.)

Kurt Gödel says:

"A set is a unity of which its elemnts are the constituents. It is a
fundamental property of the mind to comprehend multitudes into unities.
Sets are multitudes which are also unities. A multitude is the opposite
of a unity. How can anything be both a multitude and and a unity? Yet a
set is just that. It is a seemingly contradictory fact that sets exist.
It is surprising that the fact that multitudes are also unities leads
to no contradiction: this is the main fact of mathematics. Thinking [a
plurality] together seems like a triviality: and this appears to
explain why we have no contradiction. But 'many things for one' is far
from trivial.

This [fact]--that sets exist--is the main objective fact of mathematics
which we have not made in some sense: it is only the evolution of
mathematics which has led us to see this important fact. In the general
matter of universals and particulars, we do not have the merger of the
two things, many and one, to the extent that multitudes are themselves
unities. Thinking [a plurality] together may seem like a triviality.
Yet some pluralities can be thought together as unities, some cannot.
Hence, there must be something objective in the forming of unities.
Otherwise we would be able to think together in all cases.

Mathematical objects are not so directly given as physical objects.
They are something between the ideal world and the empirical world, a
limiting case and abstract. Objects are in space or close to space.
Sets are the limit case of spatiotemporal objects--either as an
analogue of construing a whole physical body as determined entirely by
its parts (so that the interconnections of the parts play no role) or
as an analogue of synthesizing various aspects to get one object, with
the difference that the interconnections of the aspects are
disregarded. Sets are quasi-spatial. They have an analogy to one and
many, as well as to a whole and its parts."

[Kurt Gödel--quoted in: Wang, Hao (1996). /A logical journey: From
Gödel to philosophy/. Cambridge, MA: The MIT Press.(p. 254)]

Regards
PH

.