In a "foundations of math" undergraduate course when I was talking about the axioms of set theory I said: "Well! The most trivial example of a set is a set which has no elements we call it the empty set and this axiom simply says that it exists in the world of mathematical objects."

Suddenly the students began to discuss on impossibility of the existence of a set with no elements and when I tried to convince them by showing the empty intersection of two disjoint sets they simply said that "OK! But this just shows that two disjoint sets have no intersection!" (In the other words they were trying to say that the intersection of two disjoint sets is not a set necessarily. Equivalently the world of mathematical objects is not closed under the intersection operator. In fact this is a reasonable statement even when we add empty set to the world of mathematical objects because $\cap \emptyset=V$ which is not a set.)

We had a similar discussion when I explained Russell's paradox on non-existence of set of all sets and I felt that it is completely unnatural theorem for students because they believe that the set of all sets trivially exists.

Question 1. What are effective ways for convincing undergraduate students that the current foundation of math is really "natural"? (If it is really natural at all!)

In fact one can build mathematics in both upward and downward directions by assuming empty set and set of all sets as the basic building block of the world respectively (it suffices to change the interpretation of $\in$ relation). Also there are axiomatic foundations of mathematics which imply the existence of the set of all sets (e.g. Quine's New Foundation $NF$)

Remark 1. Note that the upward or downward direction of building mathematical world changes our point of view about our actual world too. For example counting apples in a set of all sets - based point of view could be something like this:

All apples,

All apples minus one apple,

All apples minus two apples, (in fact "All apples minus one apple minus one apple")

All apples minus three apples, (in fact "All apples minus one apple minus one apple minus one apple")


In our current empty set-based point of view we enumerate apples of the actual world in this way:

No apples,

No apples plus one apple,

No apples plus two apples, (in fact "No apples plus one apple plus one apple")

No apples plus three apples, (in fact "No apples plus one apple plus one apple plus one apple")


Note that we don't need to define the notions "one", "two", "three", ... or even "plus" or "minus" to make the above statements meaningful. We are just using the notions of "preceding" and "succeeding" (in the sense of successor operator in Peano Arithmetic). Using this change in our thinking foundation we can look to the world in an inverse (or maybe correct) way! Living in such an inverse universe could unfold something interesting for us.

Question 2. Is there any research to show that which one of the notions of set of all sets and empty set are more natural intuitively? By research I mean a poll amongst the society of those who are not aware of the formal foundations of math including little children, usual people, students who don't work on math and math students who didn't see any axiomatic approach to foundations of math before. Something like this:

Which one is more natural?

  • (1) A set which all sets are its elements.
  • (2) A set which has no elements.
  • (3) Both.
  • (4) None.

My personal limited research shows that the popularity of the empty set amongst people is too low and amazingly the strange answers (3) and (4) are more frequent amongst little children! (It is somehow strange to live in a mathematical world which is endless in both up and down directions also it is a bit hard for me to imagine the world with an end in both up and down directions!)

Remark 2. As a set theorist I am one of the fans of set of all sets (and its philosophical consequences). Also I am interested in non-well founded set theory. The question is because I want to define an undergraduate course on "Non-well founded foundations of math" and I am interested in being familiar with students "natural" background on the subject to decide on the best approach. Fortunately our department allows me to teach "Non-well founded foundations of math" course even before "foundations of math".

Question 3. If the researches show that the natural thinking system of undergraduate students is non-well founded, is it a good idea to begin teaching the axiomatic foundations of math by a course in non-well founded set theory? (e.g. Teaching $NF$ instead of $ZF$) What about teaching some non-well founded set theory beside usual "foundation of math" course? (e.g. Teaching some basic facts about Quine's Atoms $x=\{x\}$ and $AFA$, $BAFA$, $SAFA$ anti-foundation axioms)

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    $\begingroup$ The nonexistence of a collection of all sets is a very fussy thing. It depends very strongly on the particular set theory in which one works. ZF doesn't have it, but Morse-Kelley set theory does, and I don't think anyone can make a reasonable argument that the students should somehow find the consequences of ZF more intuitively obvious than those of MK or NFU. I would suggest that if you think the nonexistence of a set of all sets is obvious or natural, it's because you don't know enough about it, and perhaps you shouldn't be hassling the students about it either. $\endgroup$ – MJD Mar 25 '14 at 1:44
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    $\begingroup$ I agree with @MJD that specific features of the axioms of ZFC should not be thought of as natural or intuitively obvious. The first version of set theory developed by Cantor included both the empty set and a universal set, and it wasn't until Russell's paradox was discovered that it became clear that a universal set might be problematic. $\endgroup$ – Jim Belk Mar 25 '14 at 1:49
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    $\begingroup$ I am curious to hear from the subtractors why they downvoted this question. $\endgroup$ – Andrej Bauer Mar 25 '14 at 8:07
  • $\begingroup$ @AndrejBauer Hello Andrej! Welcome to MESE! Thank you very much for your support. $\endgroup$ – user230 Mar 25 '14 at 8:44
  • $\begingroup$ Which is "more natural" depends on what the student has encountered before. $\endgroup$ – Confutus Mar 25 '14 at 21:31

Regarding the empty set: if your students object to its existence, and you suspect they might, set them up for a surprise. Introduce first the separation axiom, then ask whether there are any sets. Then ask if there is the empty set. When they say "no", prove that it exists by separation and the claim that there is a set. You can now pompously declare that they are illogical (hint: wear pointy ears that day). Warn them that one has to follow the logic, no matter how much it goes against their beliefs. They may question initial assumptions, but not their consequences.

Regarding the set of all sets, again, this is an opportunity to show that things are not that obvious. Let them have the set of all sets, and derive from it that you are the pope. After all, you are a saint already.

  • $\begingroup$ First of all thanks for your useful answer because of its new point of view in beginning with Separation Axiom (instead of empty set axiom) as one of the most natural properties which we expect from sets. I will test it in the future courses. I strongly believe on the point which you mentioned correctly. They should ask just on natural intuition behind axioms and after that they should follow the logic. Of course discussing about empty set axiom is legitimated by this philosophy. Regarding the "philosophical consequences of the existence of set of all sets" you got the point verrry nice!!! :-) $\endgroup$ – user230 Mar 25 '14 at 9:03
  • $\begingroup$ By the way, you don't have to do the Separation axiom before the empty set. You can propose the empty set first, and scratch it if anyone objects, but ask them whether there are any sets that they like (they will). Then you simply continue and after you've done Separation you show them that the empty set is back in the game. $\endgroup$ – Andrej Bauer Mar 25 '14 at 13:15
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    $\begingroup$ Surely, if they are willing to write off $A \cap B = \emptyset$ as "these sets have no intersection" rather than "these sets have empty intersection", then they are willing to write off $\{x \in A : 0 = 1\}$ as "this property does not specify a set" rather than "this property specifies the empty set"? $\endgroup$ – LSpice Sep 6 '14 at 6:59

Pedagogical answers:

For me, it is very natural to think of a wallet that contains no money; with this analogy my students have no problem with the empty set.

In terms of a wallet that contains not only all money but all wallets (including itself), this is an easy sell to dissuade students from walking down this road.

Mathematical answer:

There are a lot of scary things out there, so taking all of them as a default seems backwards to me. For example, I'd rather start with simple functions (polynomials) and add more complicated ones, than start with all functions (including 1, 2, and 3) and remove things.

  • $\begingroup$ For the empty set I usually use the $\emptyset=\{\}$ notation at the beginning and then describe it as a bubble which contains nothing but is not nothing because there is a thin curtain around it which gives him an entity! I describe $\{\}$ symbols by that thin curtain. $\endgroup$ – user230 Mar 25 '14 at 6:12
  • $\begingroup$ I am completely agree with you that we should begin with simple and natural theories and in the case of functions, polynomials are really natural in any sense but here my question is that which one of well-founded and non-well founded set theories are more natural? $\endgroup$ – user230 Mar 25 '14 at 6:17
  • $\begingroup$ You are asking a philosophical question that does not belong in matheducators.stackexchange.com, because it has next to no value in pedagogy. Also, the axiom of regularity is more natural than its negation; our natural experience of objects in containers excludes the possibility of a container containing itself. $\endgroup$ – Vadim Ponomarenko Mar 25 '14 at 13:29

Disclaimer: This answer uses lists, not sets, quite different objects, but I guess there are enough similarities to make it meaningful. As the notion of sets of all sets has its own problems, I will use the set of natural numbers instead, i.e. I change the question to: what is more intuitive, the empty set, or the infinite set of all the natural nubmers?

For lists, it is: what is more intuitive, the empty list, or the infinite list of all the natural numbers?

TL;DR It seems that inductive approach is more intuitive.

The process of construction (in your language building from empty set) is frequently called induction, and the deconstruction (starting from "full" sets and going down) is called coinduction.

I taught both approaches during an introduction to programming course, for example, consider lists and their duals streams.

  • $\mathrm{List}\ \alpha$, that is, the type of lists of elements of type $\alpha$, describes an inductive data structure defined by two constructors \begin{align} \mathrm{Nil} &\;: \mathrm{List}\ \alpha\\ \mathrm{Cons}&\;: \alpha \to \mathrm{List}\ \alpha \to \mathrm{List}\ \alpha \end{align}For example $l \equiv [1,2,3]$ would be $l = \mathrm{Cons}(1,\mathrm{Cons}(2,\mathrm{Cons}(3,\mathrm{Nil})))$.
  • A stream type $\mathrm{Stream}\ \alpha$ describes a coinductive data structure defined by two destructors: \begin{align} \mathrm{head} &\;: \mathrm{Stream}\ \alpha \to \alpha\\ \mathrm{tail}&\;: \mathrm{Stream}\ \alpha \to \mathrm{Stream}\ \alpha \end{align}For example the stream $s \equiv \langle 0,1,2,3,\ldots \rangle$ of natural numbers would be defined by $(\mathrm{head} \circ\mathrm{tail}^n)(s) = n$.

Each time the effect was that lists were easy to understand, while streams posed some problems. It was simple to introduce lists and how to manipulate them, while streams caused confusion and were considered "an advanced topic". Any student were able to code a program to produce list of the $n$ first natural numbers (this is inefficient, but I wanted to make the codes similar) \begin{align} &\verb`let natl n = `\\ &\quad\verb`if n == 0 then []`\\ &\quad\verb`else 0 : (map (+1) (natl (n-1)))` \end{align} but definitions like the following would take time to digest $$\verb`let nats = 0 : map (+1) nats`$$

To conclude, it seems that inductive approach is more intuitive.

I hope this helps $\ddot\smile$

  • $\begingroup$ (+1) Really interesting point of view! Existence of $\omega$ (the set of all natural numbers) and large cardinals are really as strange as existence of $\emptyset$ and set of all sets! Inspired by your answer now I am curious to know what happens if I teach the foundation of math by category theory or modal logic point of views to undergraduate students! $\ddot\smile$ $\endgroup$ – user230 Mar 25 '14 at 10:43
  • $\begingroup$ @SaintGeorg I wouldn't mix $\omega$, $\aleph_0$ and $\mathbb{N}$. Those are different concepts and might cause confusion in students (definitely not for these just starting to learn set theory). Indeed, the existence of the set of all sets would be strange for me, far more than just $\varnothing$ or $\mathbb{N}$. $\endgroup$ – dtldarek Mar 25 '14 at 11:06