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

Noah Schweber said this is a good question, which is suited for math.SE.

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.

## closed as unclear what you're asking by Henning Makholm, Shaun, Vladhagen, C. Falcon, астон вілла олоф мэллбэргFeb 1 '17 at 0:14

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• Voting to close as unclear what you're asking, since my attempt to answer you was just met by obfuscationist \word-picking and demands that I quote the parts of the question I answer -- which I had in fact already done! – Henning Makholm Jan 31 '17 at 17:56
• And in your quote I didn't talk about the purpose of a quote. Can't you argue something out without becoming emotional? – user7280899 Jan 31 '17 at 17:59
• Sorry, but life is too short to waste it on trying to help people with your attitude ... Good luck with finding someone else who will play your games. – Henning Makholm Jan 31 '17 at 18:05
• And your attitude is to insult people? – user7280899 Jan 31 '17 at 18:06
• dpmms.cam.ac.uk/~wtg10/settheory.html – Julio Maldonado Henríquez Jan 31 '17 at 18:09

There are two basic notions of what sets are.

The first is that a set is a subclass of some universe $\mathcal{U}$. With this notion of set, we do things like take intersections, unions, and complements. We might propose that some variable $u : \mathcal{U}$ is restricted to some subclass. Things like that.

The second notion of set is that of having trivial structure. The relevant thing one does with such sets is to talk about functions between them.

It's the second notion of set — that we can talk about functions between them — that is why sets are so important in a foundational role.

And it's this second notion of set for which all three-element sets are the 'same'. One can certainly impose additional structure that allows one to distinguish between different three-element sets — such as choosing embeddings of them into $\mathbb{N}$ (along with choosing a zero element and a successor operation on $\mathbb{N}$ to remember its natural number structure) — but one acknowledges that this really is additional structure above and beyond the structure of merely being a three-element set.

Of course, one can have even more complicated set notions; e.g. the idea of set one might get from ZFC by focusing on the membership relation is that sets are rigid, accessible, rooted trees. This is great kind of structure for talking about a hierarchy of sets containing sets — but it's terribly misleading if what you care about is functions between sets.

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?

No, this is not in my experience usual terminology.

In the short quote you give here, it looks like the purpose is simply to provide an intuitive motivation for the concept of cardinal numbers. And perhaps also to suggest an analogy to similar concepts from algebra (or categor theory) -- for example an "abstract group" can reasonably mean a group "up to isomorphism", and these "abstract sets" are then "sets up to bijective correspondence".

In any case, this concept of "abstract sets" is not the same as the difference between hereditary sets and set theory with urelements. For example in standard set theory without urelements, $\{\varnothing\}$ and $\{\{\varnothing\}\}$ are definitely different sets, but have the same cardinality.

• What do you mean with "it looks like the purpose is simply to provide an intuitive motivation for the concept of cardinal numbers"? – user7280899 Jan 31 '17 at 17:46
• Also, your answer just answers one question: " is the term "abstract set" usual?" You also quoted my question "What's the purpose of working with abstract sets instead of normal sets?" but you didn't answer. In my post there many more questions you don't discuss in your answer though (and don't cite). – user7280899 Jan 31 '17 at 17:48
• @user7280899: Um ... I mean just what it says. I believe is is an English sentence that expresses the meaning I intend to express. – Henning Makholm Jan 31 '17 at 17:49
• @user7280899: If you want multiple questions answered, you should not squeeze them all into a single post. The rule on this site is to make one post per actual question you want an answer to. – Henning Makholm Jan 31 '17 at 17:52
• @user7280899: The purpose of the passage whose purpose you ask about! – Henning Makholm Jan 31 '17 at 17:52

If you don't know much about the foundations of mathematics, then why are you asking these questions? At the most extreme views of formalism (which ought to be referred to as conventionalism), "the usual notion of 'set'" is meaningless outside of a formal axiomatic system. By the principle of "definition in use" the meaning of undefined primitives is wholly given by the axioms.

One of the early critics of set theory as a foundation had been Skolem. He criticized Zermelo's views with the observation that the theory could not be categorical. But, among his contributions had been that the theory of pure sets is sufficient for mathematics. A set theory with urelements is different from a set theory without urelements. Which one is "usual"? Or, is one mathematical and one not? Are you disagreeing with Skolem?

When you were reading Lawvere and Rosebrugh, did you take the time to notice expressions like "cohesive set" and "variable set"? Within their work and the broader work of a category-theoretic framework, there would seem to be distinctions that justify the use of "abstract set" without it being an Orwellian newspeak. If anything, the use of this expression is justified because the expression "cardinal number" now has connotations different from how Cantor used the term.

Returning to the matter of formalism, you give the example of $\{ 0, 1, 2 \}$ and $\{ 1, 2, 100 \}$ as if it is obvious that they are different because you have interpreted the singular terms intended to denote objects. Unless you are working in a theory where those denotations can actually be shown to be different, they are just "bags of dots". This is called the "parameterization of language". Until you actually specify a model interpreting the singular terms, you do not even know if the collections denote sets with the same cardinality.

There is a great deal of skeptical reasoning that has motivated different programs in the foundations of mathematics. One cannot simply assume that one's questions are well-construed because the subject matter does not have the kind of certainty one expects after one's initial learning experiences.