# What should one know about abstract sets and structural foundations?

Recently I came by accident across the book sets for mathematics by Lawvere. It says:

First we deplete the object of nearly all content. We could think of an idealized computer memory bank that has been erased, leaving only the pure locations (that could be filled with any new data that are relevant). The bag of pure points resulting from this process was called by Cantor a Kardinalzahl, but we will usually refer to it as an abstract set.

Note that Kardinalzahl is the german word for cardinal number. What's the purpose of working with abstract sets instead of normal sets? Also, is the term "abstract set" usual? I've seen the term "cardinal number" used more frequently.

Some basic set theory is something one learns early in one's study of mathematics. A first intuitive description might read something like this:

A set is an unordered collection of objects. If $S$ is a set and $o$ an object, then $o\in S$ is a statement which is either true or false. Two sets $S, T$ are equal iff they have the same elements. For example, $\{0, 1, 2\}$ is a set; and $\{1, 2, 100\}$ is a different set (though of course they have some common elements, but after all not exactly the same elements).

Do I understand it correctly that in Lawvere's theory of abstract sets, one identifies sets that are isomorphic (i.e. equinumerous)? I really wonder what the purpose of this is. Ask a mathematician at random whether $\{0, 1, 2\}$ and $\{1, 2, 100\}$ are the same sets.

These were my first thoughts about the topic. After that I visited the nlab and learned about the material vs. structural set theory issue in foundations of mathematics. I don't know much about the foundations of mathematics, but from my own experience I would immediately respond that normally when one says "set" one means what they call "material set" (however, in practice one might want to work with urelements, and not only pure sets). What they refer to as "abstract set" is normally called "cardinal number". So why is there this distinction? Why does Lawvere considers the category of abstract sets? I'm pretty sure that this category is equivalent (isomorphic) to the category of sets.

I know this is not about research math. But there might be people interested in foundations, so I don't see how this question could damage mathoverflow.

• I think this is a good question, but would be better suited for math.stackexchange. – Noah Schweber Dec 13 '16 at 22:37
• Just to respond to your question about {0,1,2} and {1,2,100}: it depends on whether you mean these as abstract 3-element sets, in which case they are clearly isomorphic, but not canonically, or as sets of integers, in which case there is more structure around, namely the obvious inclusion as subsets of $\mathbb{Z}$. They are clearly not isomorphic as subsets. Context is still important, even if the structure is minimal. "I'm pretty sure that this category is equivalent (isomorphic) to the category of sets." <- what is "the" category of sets? ;-) – David Roberts Dec 13 '16 at 22:47