Does equality between sets contradict the philosophy behind structural set theory?

[Leopold Schlicht <schlicht.leopold@gmail.com>]

Zermelo-Fraenkel set theory (with choice) is commonly accepted as the
standard foundation of mathematics. It is a material set theory. This
means that for every two objects/sets a,b one can ask whether a=b or
not. Also, one can always ask whether "a is an element of b" is true
or not. So "element" is a global element relation.

As an alternative foundation for set theory, Lawvere proposed ETCS (=
elementary theory of the category of sets). It is the standard example
of a structural set theory. The idea behind structural set theory is
that  elements—in contrast to material set theory—have no internal
structure, i.e. are just "abstract dots". Thus there there is no
global element relation, and "objects" are not indepedent things, they
always lie in a particular structure (for example, we cannot speak
about 2 as an object that exists on its own, but we instead talk about
the "2 in the structure IN" or the "2 in the structure IR", and
strictly speaking, these are not the "same" objects, but we have a
natural injection IN -> IR which maps the "natural number 2" to the
"real number 2").

There have been controversial discussion whether ETCS is more
appropriate as a foundation for mathematics than ZFC. I don't want to
discuss this here, but want to point you to this paper:

"Rethinking set theory" by Tom Leinster (it's available at the arxiv)

in which Tom Leinster introduces ETCS and argues that this
foundational system is much nearer to the practice of mathematicians
than ZFC.

Quite at the end of the paper, Leinster states that *Sets for
Mathematics* by Lawvere and Rosebrugh is the quite the only book that
teaches set theory in the flavour of structural set theory
(ETCS)—which makes sense, since Lawvere is the "founder" of structural
set theory.

Now, let me come to my question: Does equality between sets contradict
the philosophy behind structural set theory? Obviously, when doing set
theory in the spirit of structural set theory, we *don't need equality
between sets*. Instead, we talk about isomorphisms, which makes more
sense structurally. Also, the typical definition of equality between
sets (extensionality) can't be formulated in ETCS. Thus it seems to me
that the notion of "equality between sets" doesn't make much sense in
structural set theory. Of course, we could say that two sets are equal
if there is a bijection between them, or state that equality exists
between sets without further specifying what it does (in particular,
if we identify all isomorphic sets, whether there are infinitely many
isomorphic sets that are *not* equal, ...). But this wouldn't yield to
additional value, and is thus superfluous. Having this thoughts in
mind, I was surprised when I read the following in *Sets for
Mathematics*—the standard text book for structural set theory (on page
2!):

> Notation 1.1: The arrow notation f : A -> B just means the domain of f is  A and
the codomain of f is B, and we write dom(f) = A and cod(f) = B.

Here, the authors talk about the equality of two sets (dom(f) = A).
They also use the "big bag of morphisms"-definition of category and
not the "by pairs (A, B) of objects indexed family of hom-sets". But
in this definition, one must talk about operations dom and cod which
specify for each morphism a unique domain and codomain. But the word
"unique" here presupposes that we have a notion of equality between
objects.

Could someone from the categorical foundations of mathematics clarify
my confusion? On the nLab (see nLab page on "category") there are two
definitions of the term "category":
"With one collection of morphisms" and "With a family of collections
of morphisms",
and to me it seems the second ("With a family of collections of
morphisms") is more appropriate for structural set theory. But then,
why do Lawvere—the founder of structural set theory—and Rosebrugh use
the first one ("With one collection of morphisms") in the only book
about structural set theory?


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